I was wondering because Elrond is last living member of the house of Fingolfin and a direct descendant of Elu Thingol so does he have a claim on the High Kingship of the elves of middle earth being both the last descendant of the high king of the noldor and of the high king of the teleri? how does elven succession work? and what does it mean in the over all picture?
Such a pleasant topic to begin the new year with
Well, since the whole lineage of Elrond is quite extensive, I'll try to focus on the most pivotal and conceptual points, without just listing names and facts. Let's start with saying that Elrond's ancestry is extraordinary indeed; one of the greatest of Tolkien's lore, if I may say so. Through Eärendil's bloodline he is a descendant of the Noldorin royal family that had left Aman after the end of the Noon of Valinor, and, in particular, of Turgon (the king of Gondolin); Eärendil's father, on his side, grants the Lord of Imladris a linkage to the noble House of the Edain (the noblest among Men).
Through Elwing's bloodline (his mother), Elrond is related to Elwë, the High King of the Teleri, and, most importantly, he has the holy blood of Melian (a Maia) in his own veins. So, this incredibly legendary legacy probably makes Elrond the noblest character in the Third Age by lineage, given his connection to the greatest ones of the Elves, of Men and even to the Ainur. In light of these premises and to partly answer your question: yes, Elrond could theoretically claim the title of High King of the Elves of Middle-earth without any problem.
There are some fundamental aspects that must be considered though. First of all, there is another Elf that could claim such authoritative title, and I'm speaking about Galadriel, who is widely regarded as the mightiest Elf of the Third Age and who is legitimately the only one of the Noldorin princes that survived the exile from Aman. That said, it's never explicitly stated in the books whether an Elf-maiden may ever become High Queen of the Elves and I really doubt this (even though Galadriel does hold the title of the
de facto Queen of the Elves in the Third Age). But even if she could claim that title, both she and Elrond would never desire for that anyway: in the disenchanted Third Age and with the power of the Eldar continuously waning, it wouldn't have much sense to have another High King of the Elves again (and they wouldn't have the military capacity either), given that the main characters in charge of the World's defence are Men.
Also, I guess the two would never dare to make such choice, as a sign of profound respect towards the past High Kings of the Elves, who had to withstand much more terrible threats in the centuries before.