No such a marriage is not valid. As a woman from among people of the Book can only be married with the agreement of her guardian. And you want to hide your marriage. While Allah ask us to ask for permission for marriage. and says:
This day [all] good foods have been made lawful, and the food of those who were given the Scripture is lawful for you and your food is lawful for them. And [lawful in marriage are] chaste women from among the believers and chaste women from among those who were given the Scripture before you, when you have given them their due compensation, desiring chastity, not unlawful sexual intercourse or taking [secret] lovers. And whoever denies the faith - his work has become worthless, and he, in the Hereafter, will be among the losers. ([5:5][1]5:5)
Beside this we are only allowed to marry chaste ladies.
For details on chastity in this context see:
Can we marry non-chaste women of the book al kitaab?
And for the necessity of a guardian based on Qur'an see:
Why do only women need a wali for nikah?
Beside this you don't need an imam what you need is somebody who may write down the marriage contract. And the prophet also asked us to announce marriages so hiding them is only allowed in rare cases.
A man can marry a 2nd wife without telling his first wife, but it is not a good idea, because it may cause harm (or divorce) or might cause injustice for the off-spring in case of inheritance etc.. And even if there is no "clear" evidence or order or scholarly opinion to tell the wife and the to be wife it certainly seem to disagree with the kindness and trust which are necessary for a marriage. It is like cheating the wife by hiding an important information and would be even worse if she asked you and you lied about it.
IMHO this is hardly going along with the teachings of Islam as it sounds like secret marriage and leaves a bad taste on the spousal relationship.
Note that - secretly- chatting with this girl is a sin you must repent from first. [1]: https://legacy.quran.com/5/5
