Choose a record
There's no point destroying a perfectly good record (let alone one you like), so I recommend choosing one that's no longer playable. I chose a "Hit Explosion" record that had a pretty big gouge on one side.
Expand the record's hole
I chose a 1/4" clock movement kit because most vinyl record holes are approximately 1/4" in diameter. This size clock kit includes a clock movement that has a 1/4" spindle. Using your drill and a 5/16" drill bit, widen the record's center hole so that the spindle will more easily fit into the record.
Attach the clock movement to the record
Push (or screw) the clock movement's spindle through the record's center hole. Using the supplied hardware (normally a nut and washer), secure the clock movement to the record. You can tighten the nut with your fingers -- using a wrench is unnecessary and might crack the record.
Paint the clock hands (optional)
Most clock movement kits come with black hands and, since most vinyl records are black, you might want to paint them. I painted my clock hands orange to match the label of the record I'm using. If you don't feel like breaking out the spray paint, you can order a clock movement with silver or gold hands (though I've found the 1/4" kits to be hard to find in these colors).
Attach the clock hands
Clock movement kits vary -- I recommend checking the instructions that came with your kit to ensure that you don't break the very fragile tiny hands.
Hang up your clock
The clock movement kit I ordered (which I've linked to at the top of this guide) has an integrated hanger. This is super useful and saves a ton of time. Now you just need to hammer a nail into the wall and voila! You're finished!