I think the traditional Joker everyone is used to, is typically depicted as a guy who has grown up into adulthood and then gone crazy later on in life. It looks like this new interpretation, has him going crazy and taking on the Joker persona at a much earlier stage in life. In which case getting tattoos and platinum teeth would make sense for him. If he has rejected society and no longer cares about how people perceive him. Also the classic Joker wears a lot of suits and tends to dress like a mobster type character but this Joker seems more like a street person or scavenger character. Maybe he had to join a lot of gangs to survive and slowly climb ranks before taking over. Getting tattoos would just be part of the process.