Have you tried just tightening the chain tension? Loosen the axle nuts, pull the rear wheel tight against the chain, and re-tighten the nuts, just tight enough so the chain doesn't bind when you turn the cranks. Use a box end wrench. An adjustable will just round off the corners. You'll still have to re-tighten from time to time. Getting it right will also take some practice. The nature of the beast.

If you really need a new chain, any quality 1/2" x 1/8" chain will do. We like the SRAM PC1. A multi-speed mountain bike chain is too narrow for single speed cogs. But unless you got the bike used, it sounds like the chain is not worn enough to be the problem.