There is no in universe explanation how Aquaman got his tattoos. It was director Zack Snyder's idea of getting the character tattooed.
Regarding Aquaman's skin being bullet proof, he is not completely bulletproof. He has a denser muscle under his skin.
Director James Wan replied to this same question asked on Twitter.
Oh, bullets (or whatever else) will penetrate and break his skin. He's not a man of steel. He just has really dense mass/muscle. At least that's what Geoff Johns and I talked about
So, he is not completely bullet proof but strong enough that no bullets or steel can do harm to him. That could be a reason why he blocks bullets instead of standing up to them and repelling them.
This may not be a in universe explanation but these tattoos give a stylish look for the character Aquaman and more importantly a symbolism that he is the king of the ocean. The tattoos he has on his body are very close to ocean creatures like fish and shark.
From an analysis on Aquaman's costume from ign.com,
Momoa is Hawaiian and has said that he wants Aquaman to represent the Polynesian people, who use tattoos as a big part of their culture.
The shoulder/arm tattoo looks like fish scales, which is a nod to Aquaman's orange fish-scale shirt. The scales could actually be spearheads or shark teeth, which could speak to the deadly nature of the character
Though it was an idea of Zack Snyder, Jason Momoa has major influence in deciding what they look like and enjoyed having those faux tattoos on his body. Actor Jason Momoa comes from Polynesia. Tattoos are a part of Polynesian culture. He was so excited to include Polynesian culture in Aquaman because Aquaman is considered an embodiment of their culture. So, you see tattoos in movie version. Originally Aquaman from comics doesn't have tattoos.
There are various depictions of Auqaman in comics. He looked something like this with scaly suit. So, we can say those scaly tattoos were a nod to his costume from comics.
Or classical orange and green suit which even Jason Momoa is shown wearing in the movie.
In an interview to comicbook.com at Walker Stalker Con in Atlanta, Momoa said
“Yeah, I did go to school for Marine Biology, but the cool thing is…the greatest thing for me is that Polynesians, our gods, Kahoali, Maui, all these water gods, so it’s really cool and a honor to be playing a [water] character. And there’s not too many brown superheroes, so I’m really looking forward to representing the Polynesians, the natives.”
So, tattoos were an addition to the character to represent the oceanic culture and ocean gods.