The little pieces of plastic tubing that come on the handles of 5-gallon plastic buckets always seem to age and break, leaving bare wire handles. Replace the old tubing with sturdy PVC pipe and your handle will outlast the bucket.
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 ends of the wire handles have a sort of double bend to them. One bend takes the wire into a hole in the side of the bucket, and the other bend to the side keeps the wire from coming back out the hole.
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.