What kind of drill bit can I use for a small hole on Silestone countertop?
Silestone is natural quartz aggregate, held together by a polymer binder.  I believe it's something like 85% quartz. You'll need a diamond coated bit like this one; it's a tiny hole saw where the rim is covered in diamonds:



It's a bit tricky to start, you need to hold the drill at a slight angle so that only one side of the circumference is grinding, and then as you start to develop a crescent-shaped groove you gradually move your drill perpendicular and continue with the rest of the hole.  If you don't do this the thing will walk all over the place since it has no positive center like a regular hole saw.

For drilling upside down you'll want a spray bottle with water to keep the surface as wet as possible.  That Hitachi bit I linked to has a spiral groove going in the opposite direction of a normal drill bit, which serves to draw water into the hole, which will help you when drilling upside down.

Obviously you want to be careful not to drill too far and pop out of the top of your nice counters.

For fastening the dishwasher you'll need a plastic anchor into which to run your screws, and I've found this type works great.  In a solid material the plastic will extrude around the screw threads and hold really tightly.  They do sell them at my Lowe's but I can't find them on their website.  The 3/16" size is probably what you need for your dishwasher, and the diamond bit I linked to above is also 3/16".  Good luck!