Paint first, so you'll have less areas to cut in or tape off. You'll need to do little touch ups after the backsplash is installed most likely, but it'll be much easier than trying to paint right up to the tile.

Backsplash first. Then, once it's up, you will be able to correctly select the paint color for the area/lighting with the backsplash in place. Plus, the paint job will get messed up with the mud/grout for the tile.

I would select the paint and backsplash together (if not yet done) and then paint before installing backsplash.