Joker's also got a very strong streak of ostentatious obnoxiousness. He usually wears a suit, often tailored, but it's got garish colors and a lot of eye-catching personalization. He mixes anachronistic fashion with modern cuts, like the loopy-knot tie he sometimes wears, or elongated coat-tails on an otherwise modern tuxedo, or a man's fedora (not of that hipster, short brim BS), and occaisionally carries a cane for no other reason than its cool. His style often seems like a mockery of current fashion while still flowing together in an stylish, if perhaps madcap and utterly unfashionable, way.
Nicholson's Joker was going for a kind of artistic popinjay-gangster mix, Ledger had a Sid Vicious style look, and now Leto is going for a kind of eccentric rock star in designer duds look.
If they wanted Joker to have any tattoos, I think it's better for them to be so over-the-top and prevalent instead of just a few on his arms or chest. Jim Lee and Frank Miller's Joker is ASBAR had a gnarly back tattoo of a dragon, but it could be concealed with a suit jacket and just kind of seemed pointless. Here, it fits with the overall design for the character. Yeah, it's kind of ugly and seems stupid, but so does an old school bow tie or pimp hat if your costume doesn't fit, and I think they've arranged Leto's look so that the tattoos are an important feature.