that's that ugly rubbery moto mold crap.

cut it off and use fizik microtex.

if you've never applied new tape before, do research the method first. there are many ways. i favor beginning at the ends, taping outward, doing a figure-8 around the lever clamp, and finishing with something elegant like embroidery thread (or electrical tape cut in half).

if you found some replacement levers without the suicide levers (about $5 at a co-op), you could add cane creek gum hoods for about $9 shipped.

this is a good time to replace cables and housing if needed. but you'll need bike specific cable cutters like those made by park tools.