Just overwriting once with all zeroes (or anything else) is enough; for modern drives (where “modern” means “last 25 years”). It’s basically impossible to recover data after that.
One of these days though I'll get around to zeroing out a drive and then running Autopsy on it for shiggles just to see what comes back up. Should be a fun experience!
I recommend overwriting with an endless loop of the freeway scene from "Solaris".