My group are in the process of switching to 5e and one of my players has asked me to ret con the last couple of games so he can make a new character with the homebrew "Blood Hunter" class.
I regularly give my players an opportunity to do a "character rebuild" - the only thing I ask is that they keep the core concept of the character intact, but they can change almost anything else. If we were switching editions, I would definitely allow this. (That said, I never switch editions in the middle of a campaign, so it's a moot point...)
But if the "Blood Hunter" class can't be made to fit with the character's previous "core concept", I would also say "no".
I tell him no and he asks if he can just kill himself to make a new character, i told him no...
I think this was a mistake - instead of saying "no", I think you should have said, "no, but you can retire this character to bring in a new one".
IMO, there's no benefit in trying to force a player to continue with a character he really doesn't like, and there's certainly no benefit in a rule that only character death allows a switch. If that character's story is done, write him out and bring in the replacement.
Having said all that...
One of my rules is that players are expected to play "in good faith". That means not disrupting the game for everyone else, and in particular it means not suiciding a character just because things didn't work out for you. (This usually applies in the case where the player chose to roll his ability scores and then rolled poorly. The deal there is that
if you choose to roll, you also choose to take the results of that roll.)
A player who violates that is almost certainly revealing himself as a problem player, in which case he would find himself uninvited from future games. (Though that's very much a nuclear option, and not one I've ever had to use. Peer pressure generally deals with such things, as does the understanding around the table that we're all adults and expected to act accordingly.)