I think Ray Haffner has the tattoo, and Ray Haffner killed the private investigator (nicely played by Beth Riesgraf aka Parker from LEVERAGE, as PJ said), but Haffner isn't RJ. Here's my explanation.
The real RJ hires a private investigator, but that's not Kyra (sorry, not sure about the name). That's Ray. Meanwhile Kyra is actually hired by Tyger-Tyger. So, Kyra is hired to spy on CBI, and Ray is hired to spy on Kyra (as Cooper told, he's busy with another job).
About Kyra:
As we saw, Tygers were trying to persuade somebody within Lisbon's team to work with Tygers. That's why the guy who passed some documents to Rigsby said what he said. And that's why Kyra was trying to get closer to Cho.
About Ray:
Ray knows about Tyger-Tyger, but he is not with them. He is against them and that's why he left CBI without any serious reason. Ray hates cops and he has a tattoo, which means "death to cops".
About RJ:
RJ knows what we've presumed about Kyra and Ray. RJ hires Ray to spy on Kyra and get close relations with her, telling she's with Tygers and she's spying on Lisbon's team. Ray even warns Lisbon and Van Pelt about it. RJ persuades Ray to kill Kyra, and voila!!! Everything worked perfectly, as RJ planned. Now PJ has a false clue. Surely he'll find Ray's tattoo and that is exactly what RJ expects.
As an additional thought, Kyra's behavior was changed after she noticed something about Van Pelt. It could be some sign connected to visualists. So, if Kyra was working for Tygers, then visualists and Tygers at least dislike each other.