Possible to paint kitchen backsplash tiles?

It's hard to say without knowing more about the tile. If it has a shinny finish I would use sand paper to rough up the finish (probably do this with any and toughly clean) and use paint recommended for tile.

Take a look at this one.

http://www.11magnolialane.com/2013/03/12/painted-tile-backsplash-in-my-kitchen-a-year-later/

She is showing it 18 months after finishing the job so we could see how it's holding up, but there are links there to her original post, and what she did is also on this page. She is using oil-based paint, but I've read where people are successful with latex paint. However, the latex should be sealed to preserve the finish longer. You don't mention if you'd like to go lighter or darker but either way, prep is the most important step. Be sure to clean with a degreaser, scuff it up a bit (I'd wipe it down with a tack cloth too), and use a primer. Some of the steps may seem like just more work, but they can make the difference between a great job and one where the paint starts peeling. Also sealing your work is important - especially in the kitchen where a backsplash will be washed multiple times. Best of luck with your project. I hope you enjoy your results.

It doesn't stick well to tiles especially if there is water splashed on it.