The closest I got to the main tattoo was an alchemical symbol for amalgam, usually of mercury and silver, but in that case it would be inverted. Would inverted amalgam mean 'put asunder'? It could also be half of the symbol for Sirius, if oddly twisted, and the small one above it had the meaning of 'odelman', which I suppose could be some kind of on-set humour.
The ass-rune on his belly totally means that he bottoms, even though that's not what it symbolizes at all. But wouldn't it be funny if it did.
But the most curious thing about them, to me, is that there's writing around the main symbol; and while I have no idea what it says, it looks similar to how the Marauder's Map was laid out in the movie. Which made me think that it's actually a spell conducted in a similar fashion. But what spell? And would he have gotten them before being sent to Azkaban, or inked all of it himself while in the prison? Since he had been involved in making the Map, he certainly would have known how to do something like that.
But then again, he's got his prisoner number on his neck, so it could also mean that they were inked by the Ministry. If it's binding or a tracking spell, how would he have gotten around it? And since none of this is even hinted at in book-canon, why am I even theorizing on it?
Ps. Some of it has to say 'Cuaron ships teh puppies' in ancient Accadian. Or maybe 'I am Mulciber's prison bitch'.