Okay, I've made a start, but there's work to be done if you want to do it how you want.
After reading documentation about how the Genesis's VDP works, I found that BG A and B start at $C000 and $E000 respectively in VRAM. I then looked in VRAM in Bizhawk and found Axel's name, and searched the ROM for the same bytes I saw (they were 16-bit values - two bytes - so I cut off the high byte as I assumed they wouldn't be in the ROM). This led me to find Axel's name on the selection screen at $72032 in the ROM. Beware that if you want to use a five-letter name (like Blaze) we'll have to find the instruction that tells the game to look there and change it to look elsewhere, then put the new name in that new place.
I used the same method, and discovered the name in the opening crawl at $3FCED - the table file is easy as 00 is a space and 01 upwards is the alphabet.
The name in the bio wasn't so easy to locate, but I just assumed the ROM would use the same table mentioned above for it, and it certainly did.
You can find it at $3FA03.
I assume that's the only three locations of Axel's name in the game (unless he's mentioned in the ending, but that'll be easy enough to find). If you want to replace his name with one of the same length, now you can!
(just remember to use Fix CheckSum if you're going to go hacking Genesis ROMs) But if you want his name to be longer, we'll need a bit more work.
Fix CheckSum: http://www.romhacking.net/utilities/342/