Drop sets have always been a staple bodybuilding tool, but you don't see them used often with dips. It just takes too long to deload a dipping belt, plus for many new (or kinda fat) lifters, body weight is already heavy and reps are too low to accumulate enough time under tension. Here's a way to get around those issues.
Hold a dumbbell between your feet and perform 8-12 reps of chest dips. Choose a dumbbell heavy enough so that you reach failure or close to it somewhere in that rep range.
Once you can't do another rep in good form, open your feet and drop the dumbbell. With no rest, continue to rep out until you again reach failure. (Or leave one rep "in the hole" if you're struggling with form.)
Now, with as little rest as possible, have a partner make a platform with his hands and place your ankles or the top of your feet on his hands. He isn't going to lift your legs, rather, you're going to push off of him as needed until you can no longer do another rep in good form.
Rest for a couple of minutes, then repeat once or twice. Be prepared to yelp in pain when steering the car the next day.
Options
If you have access to Olympic lifting plates, these can replace the dumbbell. Just hold the plate between your thighs or knees and let it go at the bottom of the rep.
If you don't have a trusty training partner, the first two steps will still light up your chest.