Yes, that's the cornerstone of Shia belief.
Some Sunnis deny that Ali (as/ra) had any claim to the caliphate against Abu Bakr but others who can't deny the recorded protests by Ali (as) tend to highlight Ali's final retreat and allegiance to Abu Bakr, suggesting that the difference was a minor thing that was finally settled in a brotherly fashion. They argue that if Ali deeply believed that the caliphate solely belonged to him, he would have fought back.
Shia reject that view and believe the issue had been of profound importance and that the Imam's restraint and final retreat was not an endorsement of the legitimacy of Abu Bakr but to prevent further isolation of Ali and the Banu Hashim as well as avoiding infighting among the companions especially when there were waves of apostasies outside Medina which threatened the integrity of the newly founded Ummah. So Imam chose between the lesser of the two evils: infighting among companions and destruction of the Ummah as a result and coping with a Muslim caliphate that was not legitimate.
We have several narrations in which Ali (as/ra) highlights his superior right to the caliphate against Abu Bakr and the other two caliphs who succeeded him. These narrations are mostly from secondary Sunni sources and from Shia sources. This is for two reasons: First, Ali and his family are highly under-reported by Sunni hadith collectors whose scholarship reflected the anti-Ali and/or anti-Shia biases of their times. That's because for Sunni scholars, to quote narrations/hadiths that gave weight to the Shia view was an unforgivable political offense at the time as the Shia Imams were mostly isolated and their Shias (followers) violently repressed for almost the entire length of the Abbassid dynasty when the first hadith and historical sources were compiled by the Sunni Imams.
In this answer I provide a selection of authentic narrations from various sources for Ali's view.
I start with a mainstream Sunni source, Bukhari, which proves that there was some serious disagreement over the caliphate. This one only briefly mentions two related disputes between Ali and his family and the caliphate: the dispute over Ali's wife, Fatima's claim to the land of Fadak and Ali's dispute over the caliphate. I quote a select part from the long narration:
... So Abu Bakr refused to give anything of that to Fatima. So she became
angry with Abu Bakr and kept away from him, and did not task to him
till she died. She remained alive for six months after the death of
the Prophet. When she died, her husband Ali, buried her at night
without informing Abu Bakr and he said the funeral prayer by himself.
When Fatima was alive, the people used to respect Ali much, but after
her death, `Ali noticed a change in the people's attitude towards him.
So Ali sought reconciliation with Abu Bakr and gave him an oath of
allegiance. Ali had not given the oath of allegiance during those
months (i.e. the period between the Prophet's death and Fatima's
death). Ali sent someone to Abu Bakr saying, "Come to us, but let
nobody come with you," as he disliked that Umar should come, Umar
said (to Abu Bakr), "No, by Allah, you shall not enter upon them alone"
Abu Bakr said, "What do you think they will do to me? By Allah, I
will go to them' So Abu Bakr entered upon them, and then Ali uttered
Tashah-hud and said (to Abu Bakr), "We know well your merit [note: merit is a more accurate translation for fadhl than superiority I beleive, especially in this context. See comments] and
what Allah has given you, and we are not jealous of the good that
Allah has bestowed upon you, but you imposed on (or tyrannized) [translation correction. See comments] us in the
question of the rule and we thought that we have got a right in it
because of our near relationship to Allah's Messenger (ﷺ)." (Bukhari 64/278)
There are a lot of important points and clues in the above narration. First, the indication that the disputes were so bitter that it led to a deep grudge on the side of Fatima (as), Ali's wife and the Prophet's daughter, which is a separate issue in its own right that I don't want to get to.
Second, Ali's full isolation after the passing of Fatima is explicitly mentioned which explains Ali's final decision to reconcile with Abu Bakr and his subsequent reconciliatory tone despite still highlighting the reason for his initial opposition to Abu Bakr: i.e. Abu Bakr "tyrannized" them over a matter in which he thought they, that is the Banu Hashim, "had a share in."
Third, there's a hint that shows that even when reconciling with Abu Bakr, Ali still preserved a deep grudge against Umar. I don't know how Sunnis explain this. The Shia point to records of Umar's attack on Ali's house in which Fatima was reportedly fatally injured. Sunnis reject the reports of Fatima's injury with some agreeing that there was only a verbal threat by Umar to burn the house of Ali and Fatima, should they not accept Abu Bakr as caliph.
Now we go the non-mainstream Sunni sources which provide more instances of Ali's view and in greater details.
This long narration from Sharh-u Nahji l-Balagha (Commentary on Nahju l-Balagha) by Ibn Ab il-Hadid*, a Mu’tazilite scholar of the 7th century captures the very first dispute between Ali and Saqifa faction involving Abu Bakr, Umar and the Ansar, when the group met Ali in his house urging him to pay allegiance to Abu Bakr:
Ali said: “I am servant of Allah and brother of the Messenger of Allah (SAWA)”, until they reached Abu Bakr and told [Ali] to pay allegiance. He said: “I’m more deserving of this matter than you. I won’t pay allegiance to you. You are more deserving to pay allegiance to me. You took this matter [allegiance] from the Ansar by arguing to them from your closeness to the Messenger of Allah (SAWA) and they paid you subservience and granted you the rule, and I am more rightful according to your argument for the Ansar [for Imam Ali was even closer to the Messenger (SAWA) than Abu Bakr]. And be fair if you fear Allah from your selves, and recognize for us [Bani Hashim] from the matter what the Ansar recognized for you. Otherwise you have wronged them while knowing it.”
Umar said: “You are not left over if you pay allegiance.”
Ali said to him: “Milk it, O Umar, for half of it is yours! Support his cause today and tomorrow it [the caliphate] will pass to you! By Allah I won’t accept your saying and won’t pay allegiance!”
Abu Bakr said to him: “If you don’t give me allegiance I won’t be annoyed by you.”
Abu Ubayda told him: “O Father of Hassan! You are young and these are the elderly of the Quraish, your tribe. You do not have their experience and knowledge of the affairs. And I don’t see Abu Bakr but more capable and enduring of this affair than you. So grant him this matter and land it to him. If you fare and live long, you are deserving of this matter and rightful in your virtue and closeness and career and jihad.”
Ali said: “O group of Ansar! Swear to Allah don’t push out the dominion of Muhammad (SAWA) from his house to yours and don’t depose his family from their position among the people. Swear to Allah, O group of Ansar! This is because we the people of the house (Ahl al-Bayt) are more rightful for this matter as from us are the reciters of the Book of Allah and the perceptive of the religion of Allah and the knowledgeable of the tradition and the discreet in affairs of the people; and swear to Allah that it (rule) is within us. Hence, don’t follow your whim as you will tread away from the right!”
Bashir bin Sa’d said: “Had these words been heard by Ansar, Ali, before they sold them to Abu Bakr, no two would have disagreed about you, but they pledged allegiance.”
Then, Ali went to his house and did not pledge allegiance and stayed in his house until Fatima died and he pledged allegiance. (Sharh., pp 11 and 12)
This narrations clearly shows how serious and confident Ali was in his superior right to the caliphate and the illegitimacy of the Saqifa outcome. Therefore only the bitter incidents that reportedly followed this refusal, involving violence against Ahl al-Bayt, can explain both why Ali first didn't and finally did pay allegiance after Fatima's death.
Here's another narration from Manaqibu l-Imam Amiri l-Mu'minin (Virtues of the Commander of the Faithful) by Akhtab Kharazmi, a Shafi'i scholar, quoting a debate that took place during the counsil that was held for appointment of Uthman. Ali in his turn says:
"People paid allegiance to Abu Bakr and I am, by Allah, more worthy and rightful for it, but I listened and obeyed fearing that people turn back to disbelief, hitting one another's neck by sword. Then Abu Bakr took allegiance for Umar, and I am by God more worthy of the matter than him (Umar), but I listened and obeyed, fearing people turn back to disbelief. Then you want to pay allegiance to Uthman and I don't listen and obey, as Umar put me among five individuals and I am the sixth of them. By Allah, he doesn’t recognize for me a merit in decision-making, and they don’t recognize it for me as if we are all equal in this matter. By Allah, if I were to debate, no Arabs from them and no Persian from them and no allies from them and no pagan would be able to reject anything of it. Then he said, "I took you to testify oh five men. Is from you a brother of Messenger of Allah other than me?" They said: "No." Is from you..." [Ali's mentions several of his special relations with the prophet (as) and his services with companions all acknowledging each]. (Manaqib 1/314)
Some Sunni scholars have rejected this narration but without citing any reason for their rejection other than it contradicting their dogma. Some others have quoted parts of it without objection.
This narration confirms what was said about the reason for Ali’s compliance with the caliphs despite believing he was more qualified for the position in each case and his complain about why they treat him as equal to themselves. The testimonies that he extracts from the council on his virtues are also remarkable.
This important debate is also recorded with some variation in Ibn abi l-Hadid’s Commentary on Nahju l-Balagha which excludes Ali’s critical remarks on Abu Bakr and Umar’s caliphate but in addition to the testimonies it includes conversations that followed the testimonies. In it the five people present at Shura insist that Ali despite his unique virtues must agree to Uthman claiming that people wouldn’t accept Ali as caliph and that they would kill anyone who refuses to pay allegiance to Uthman! So Ali has to comply again against his convictions to avoid bloody conflict!
The testimony part of this debate is also recorded in Shia sources such as in al-Amali of Sheikh Saduq.
My final instance is from Nahju l-Balagha itself, a compilation of Ali's statements (as/ra) by Razi, a Shia scholar. This is a sermon by Imam Ali when he addresses people of Kufa during his reign as the fourth caliph:
Beware! By Allah, the son of Abu Quhafah (Abu Bakr) dressed himself with it (the caliphate) and he certainly knew that my position in relation to it was the same as the position of the axis in relation to the hand-mill. The flood water flows down from me and the bird cannot fly up to me. I put a curtain against the caliphate and kept myself detached from it. Then I began to think whether I should assault or endure calmly the blinding darkness of tribulations wherein the grown up are made feeble and the young grow old and the true believer acts under strain till he meets Allah (on his death).I found that endurance thereon was wiser. So I adopted patience although there was pricking in the eye and suffocation (of mortification) in the throat. I watched the plundering of my inheritance till the first one went his way but handed over the Caliphate to Ibn al-Khattab after himself… till when he went his way (of death) he put the matter (of Caliphate) in a group and regarded me to be one of them. But good Heavens! What had I to do with this “counsil”? Where was any doubt about me with regard to the first of them that I was now considered akin to these ones? But I remained low when they were low and flew high when they flew high… (Nahju l-Balagha, Sermon 3)
I didn't quote this sermon in full and added an ellipsis for brevity but do check the source and read the whole thing. This Shia narration is an important one. Here Ali's view is again totally consistent with what we read from other sources. But here he also highlights the pain and difficulty he had to endure seeing the injustices committed by the earlier caliphs.
Notes
*. The English Wikipedia's entry on Ibn Abi l-Hadid is biased and selective. I recommend this Arabic entry from Wikishia for a balanced view of his views on Imam Ali. He acknowledges that Ali was superior to other companions in merit but says a caliph doesn't have to be the best available person, therefore rejecting the Shia view.
