How do I install an adhesive backsplash in the kitchen?

For a peel and stick option, you just need to make sure it’s clean (use a wall cleaner) and make sure it’s dry. If you’ve repainted recently, give it a few days to ensure the paint is cured.

Peel and stick tiles are a great option. There's a wide choice available at your local home improvement store and/or online. Be sure to follow the instructions.

Hopefully you don't have flat paint on your walls. Most people don't in a kitchen but you never know! It would be hard to ever remove peel and stick tiles from flat paint. I would make sure that the paint was a satin or semi gloss finish for easier removal later.

Make sure the walls where you are going to stick the backsplash are even. I don't mean exactly level just not hard bumps or curves because it will be really easy to see when you put the backsplash up.

It will depend on the your current wall as to whether you need to do anything other than just clean to add the peel & stick. Painted surfaces need to be painted well enough so that if you remove them later it won't destroy your walls. If you have any texture, you will need to even it out of the texture/grooves will show through after a while. Example, if you are going over tile or paneling.

I would clean with a degreaser before I put them up so they adhere well.

To reply to your question about painting over the flat paint with a semi gloss or a stain. I sure would. Any time you adhere anything like wall paper or self adhesive tiles to a flat painted wall it makes it harder to remove them later. You don't have to paint the whole kitchen if you don't want to. Just the areas where you plan on using the peel and stick tiles.

Be sure to use an adhesive primer. It’ll make adhering easier and if you ever need to take it off, removal will be easier and less damaging.

This method is incredibly easy: https://bachelorettepadflip.com/2014/03/03/diy-kitchen-backsplash-tutorial/