What is a good paint to paint brick with for a back splash?

A water based acrylic will work but you might need a primer if you are going from a dark brick to a light colored paint.

Not for a backsplash you don't. What a nightmare. Id use exterior latex. Tough enough to scrub

Get a brick stain instead of a paint.

Brick and Tile Paint........or a Brick Sealer and or Gloss paint???

You could go with enamel and a polyurethane sealer for shine. The bricks will look something like subway tile, and you can paint the grout any color you want. Be sure to caulk and otherwise seal any seams so water won’t fall behind your sink. Sounds interesting. Please photograph your progress from before to after, and write it up for a Post on Hometalk! ☺️

Be sure the brick is clean and apply a good primer before painting the color of your choice. You could even use more than one color....maybe paint the majority of the bricks a neutral color then adding in a few colors here and there to coordinate with your home's color scheme in a random pattern. Good luck and would love to see what you do.