Depends on the character, I'll use what I'm currently working on as an example

On the right is the character I'm theming around, on the left is the graphic I'm making, and in the middle is the gradient map I'm using
The colours I choose for the gradient map are based off of the colours of the character, so since he's purple/blueish I add purple and blue, then I also add another colour, it is either the opposite from their main colour (Complementary), a hue shift from their main colour (Analogous), or the character's secondary colour, I usually use the character's secondary colour or a complementary colour while subtly hue shifting towards that colour because I like the contrast in my themes and graphics (I am an artist I've been taught to make sure the colour pallete of my art is interesting), so here I chose a light pinkish/red orange and hue shifted from purple, to pink in the darker areas so it isn't too harsh of a colour change
Then I'd colour pick the colours from certain areas like the most prominent colour and test them around on my theme until it looks good, an example I have of this that's still up is @reader, though for this I did lower the opacity of the gradient map layer to 50% so you can still see the original colours a bit
And that's my process on colours for graphics and/or themes, I hope it's helpful in some way or just interesting to read about haha