I use the old buckets mostly for working with cement. A good handle makes carrying the heavy buckets a lot more enjoyable.
Step 1: Removing the Handles
The wire is springy. To forcefully remove the wire you need to grab what you can of the wire at the first bend with a pair of pliers and give the wire a twist. That lets you slide the end of the wire back out the hole.
To get what's left of the old plastic off, beat it with a hammer, cut it with snips, or use a hammer and chisel. Sometimes you can just rip it apart with pliers.