I have a couple problems with this, I keep seeing people mention "context" possibly making the tattoo's "better", I disagree. First, no amount of context, whether it be shallow or deep reasoning, will ever make me like the face tattoo's. They're just ridiculous and completely unnecessary no matter what the explanation is. I've seen tons of theories on the damaged tattoo and they're all just as pointless as the tattoo itself.
But even more importantly, if they spend time trying to explain the Jokers tattoo's, then something is wrong to me. That worries me about the writing of this movie, it's a completely pointless storyline that really adds nothing and is completely unnecessary. All those reasons you just speculated on, well, I don't care or think they should waste time on something like that. There are much better things to focus on. Outside of the possible Robin tattoo, which could be some sort of a reminder/trophy for the Joker's killing of one of Batman's sidekicks, there is no reason they should be wasting any time explaining tattoo's. The rest of them are so dumb that they don't deserve an explanation. If the guy has tattoo's, he has tattoo's, I don't freaking care why. In the TV show "Prison Break" explaining the tattoo's made sense, for the Robin tattoo, spend a small scene explaining that, fine, that makes sense but the rest of the tats, there is no reason I can think of that's interesting enough to waste time on them. I don't need context for something that's so pointless to begin with.
Maybe he got the tattoo's to make fun of prison culture, or because he thinks they're funny, or because he's in love with himself, or because Harley likes to doodle, guess what, don't care. It's a stupid plot point they shouldn't waste time on. Giving the Joker tattoo's doesn't make the character more visually interesting to me and to top it off, the tattoo's they gave him are stupid and lack purpose (again, outside of the Robin tat). The character of the Joker has a vast amount of interesting details to expand on without adding something as pointless as why he has tattoo's.