You probably don't need to purchase a new crankset.
You can remove the the rings you don't need and simply add a new ring. You'll need to know the distance of the bolt holes of your current chainrings/crank. This is known as the BCD (bolt circle diameter) and you can usually find it stamped on the chainring. Common sizes include 130mm, 110mm, 94mm, etc.
Ideally, you'd want a chainring designed for single speed use. Rings designed for multi chainring drivetrains have pins, ramps and cut away teeth to help the chain smoothly move from one ring to another. Since the chain isn't moving, you don't need them.
You will also need single speed chainring bolts which are shorter than double ring bolts.
You may or may not need a new bottom bracket. In a perfect world you want the chain to be in a straight line to the middle cog in back. If you mount your single ring in the middle position of a triple ring crankset you'll probably be good to go. If you decide to go with a dedicated single ring crankset you'll likely need a new BB with a shorter spindle length.
Chain needs are more a function of the rear cogs/derailleur than the front. Any chain made for a 7,8 or 9 speed drivetrain will work fine.