In short, yes it is possible but would cost more than it's worth for the conversion because of your frame. You need to be able to tension the chain on a fixed gear, but since your frame has vertical dropouts it has no way to adjust the chain tension, and therefore would need a hub with an eccentric axle to tension the chain reliably. These hubs are a bit expensive, so you shouldn't consider it worthwhile to put such an expensive component on an otherwise inexpensive bike. Go buy a proper-fitting old steel frame with horizontal dropouts to cut your teeth on.

And read this: http://sheldonbrown.com/fixed.html