The only thing you really need to do to create a lathe from your drill press is make a live center, and that's just a piece that holds the wood and rotates freely. Here's what I did.
I ripped apart an old battery drill, a cheap skil drill from the thrift store for a dollar, Out of the pieces you should be able to save a chuck with shaft and a thrust bushing.
Now bolt a scrap of hardwood to your drill press table and bore a hole through the wood slightly smaller than the shaft of the drill chuck. For example, the chuck shaft was slightly over 5/16th of an inch so I bored a 5/16ths inch hole. Here i'm using a scrap of Mahogany.