Most counters have a backsplash independent of the counter applied using adhesives. If this is the case, removal and replacement is possible. A backsplash behind a stove may be more difficult to remove, but the solutions listed below have been used.
Paint is the least noble of all materials. That's why it is the first low-cost option for redecoration. Durability is its least favorable characteristic -- worth considering before use in this location. If this is your principal residence and will be for the next few years at least, you might also consider facing with synthetic stone (i.e., Corian and other brands), stainless-steel sheet metal, or tile, in order of quality.
Synthetic stone can be easily cut to size for vertical facing, and sill, if you have one, and can be sealed to adjacent surfaces with silicone sealants. Minimal joints and easy to clean. Sheet metal can also be cut to size, but requires special attention to form-finished edges, and can then also be finished with sealant. Again, minimal joints and easy to clean.
Tile is a traditional option and can be applied with organic adhesives, joints grouted with unsanded grout, and again sealant at the edges adjacent to other materials. Textures and joints are less easy to clean, but manageable.
If you continue on the path of paint, use a primer specified for application to plastic laminate, and a high-quality glossy paint.
For more information:
Read more about green countertop materials in GreenHomeGuide's Countertop Know-How section.