For me, maybe Razor from Maniac Mansion, even though I never made it more than ~30 mins into that game. My kid brain didn't understand the funny tingly feeling but years later I realized, "oh duh, hot punk babe".

There was also Princess What's-Her-Name, though she wasn't much of a character.

In terms of characters with actual detailed personality, the first big one was probably Rinoa from FF VIII. I was the same age as her when I played it.