To get your hands dirty means to do some sort of labour. Literally it would just mean getting your hands dirty, but as a figure of speech it can mean, for example, cleaning. If you were talking about something illegal though, think of mafia bosses. They have their hired killers so that they don't have to 'get their hands dirty' by killing people. They don't want to get physically involved with it, so somebody else does their dirty work.

It's plural: "get one's hands dirty". It basically means you do something illegal. Which doesn't look good. Like dirty hands. "Dirty work" refers to unpleasant or mundane work, but can also mean illegal jobs. Not sure about "dirty tips"?