You know, it's different for about all the characters I've had.
Back when I first made characters, like back in my RPing youngun days, I'd look for pictures that could be RP characters, and once I found one I liked I'd put a personality and a name to it. Sometimes with the pictures I'd see stuff and think, "What could be the reason for that?" and it would end up being a big role for my character.
I still sometimes make characters that way. I'll play around on Picrew and like the design so much I want to keep it and add to it. But lately I make characters more by starting with a concept or a name, both usually start as jokes but I like building on them once I have a face to match. For concepts, they tend to come up randomly? Just a random thought that I'll write down and want to keep for future use. And sometimes it's to try some possible plot holes I have with other characters.
Lots of times, both in my youngun and current days, whenever I start writing out the character I'll get lots of ideas right then and there. There have been times when I've had ideas that I couldn't really use, but after time I'll have a character and think, "Hey, I could add that tidbit to them!!" because they have enough stuff to not need to rely on that sole trait. It's happened a lot more recently, with Mori's weapon or Hopper's worries. And how I mentioned that ideas come while writing, I was writing a reaper character about a month ago and on the spot made up something that could explain their existence, but lately I've really liked that idea I made up and want to blow it into its own world, kinda deal. I guess that kinda connects with my thought process; you know myths where it explains things like why crows are black, why there are eclipses, things like that? I've always loved those. The idea that something spectacular and paranormal is the cause for stuff we see all the time has always been fascinating to me, so I like trying to mix that with my writing, with the whole modern/fantasy combinations. So I think it bleeds into a lot of my thought process.
I also do tend to think of my other characters a lot when making new ones. Most of the time they're characters that are built revolving around a current one, but sometimes when I have a brand spanking new one I like to think about which of my current ones would get along with them. You can tell with all the links I have KEKEKEK. And no matter how wild or how different my characters look from one another, they all belong to the same world! So it's bound to have fun interactions.
In terms of avoiding doing the same thing, I don't really worry about that. If anything, I like to go the opposite direction. Of course I don't want to make carbon copies of personalities and such, but I really enjoy taking overused tropes and archtypes and kinda building off it. A lot of my characters are built to be supporting roles for protagonists, so I have this mindset of "Let them be normal so that protag can stand out".