Taking the question at face value, the best way to clean mucky, but not kitchen-grease-soaked terracotta, is to pre-wet with water, then get some of Kevin's Xtreme clean down (start at 1: 10, but try stronger if not happy), leave a few minutes, scrub in with a soft-ish brush on a medium weight slow rotary, north/south then east/west, then power rinse off with a truck mount and a spinner, start at 500 psi, and work up until it's high enough. Resort to solvents if you have to, but try Xtreme clean neat, first.
The real question is, what's the best way to seal?