Q:

What is the story behind the Thug Life tattoo?

In one Futurama episode, Farnsworth makes a remark about Fry's small Bender tattoo, just saying it's a little ink.  Then Farnsworth left showing a large tattoo of the words "Thug Life". In other episodes (especially the 2 min documentary on his life) Farnsworth was never in a gang.
What is the story behind that tattoo?

A:

There is never any reference of the professor being in a gang in any episodes of Futurama, only in the movie Bender's Big Score. To further back up my statement, we have seen Farnsworth naked several times. Farnsworth having the tattoo was likely just meant for laughs.
However, we could (and should) see an episode explaining more of the professor's past, including his alleged gang life. We have only been able to glean bits and pieces of information from his life from episodes such as "A Clone of My Own" where we learned small tidbits of his childhood, so there could certainly be several episodes in the future devoted to shedding light on the professor's past. 

A:

The thug life tattoo is an often seen joke (typically displayed on an otherwise unassuming or mild-mannered character) to hint at there being a somewhat darker side to them. This seems to draw from the famous hip-hop artist Tupac Shakur, whose trademark stomach tattoo read "Thug Life"

This can be compared to Farnsworth's tattoo, which while on his back rather than his stomach, is of a very similar style.

While the tattoo's meaning is claimed to be an anagram for a more social message, it would also appear to extoll the virtues of the modern gangster lifestyle. The joke arises from the juxtaposition of this message with the personality of the character on whom it is emblazoned.

