Sorry for the grammatical mistakes. English is not my first language.
Recently, I have fallen in love with a girl and I don't want to do an unlawful relation. Instead, I want to directly marry her. The problem is, she is my relative. She is my cousin's daughter (father's sister's daughter's daughter). I am half year older then her. Is it permissible for me to marry her?
Yes you can as she's not a close relative of yours.
Your cousin is not considered as a close relative and you're not considered her mahram. Therefore, she is still a woman who needs to wear hijab in front of you and cannot be alone with you.
This proves that her daughter is also not a close relative of yours and is like a stranger, even though you have blood ties.
The evidence for this is that there's no ayah or sahih hadith forbidding you from doing so.
Muhammed , صلى الله عليه وسلم , married to his cousine Zaynab; and cousin of Muhammad, Ali, married to daughter of Muhammad, Fatima.
It is said in Quran 4:23 that it is not allowed to marry to sisters. But there is not said that relatives that are far / not close at some level , may marry, so, probably the word meaning "sisters" mean only most close / true sisters (with same/mutual mother or father or both). Additional proof is that "sister" does not mean "cousin sister" in Arabic, I do not know that well, but Google Translate says "cousin" is "ibn gam" ie they use phrases like "uncle's daughter" for them instead of "sister". Also an additonal weak proof is that it is said that aunts are not allowed and daughters of sisters or brothers are not allowed, but it is not said that daughters of aunts or uncles are not allowed, but it could be said, like daughters of sisters & brothers are said, though, is "sister" mean "cousin sister" they would be already mentioned and indeed not needed to mention second time. Also an additional weak proof, if Arabic has different words for aunts, it is somewhat expected that aunts' daughters cannot be signed with same word as sisters, while they are even more far relatives than aunts, though that happens in Russian language for example, where special word for uncle but sister may mean cousin sister. But I think they at least once say "cousin" or "cousin sister" in Russian and only then shortly say "sister", and do not say "(my) sister, you ..." to her, only say about she to other persons "my sister, she ...". (And I am not very sure).
