Providence is the Trust, the organisation is are the true power behind the country (world in Providence's case). The Shadow Client is Agent Graves (no relation to Penelope Graves), he gives people proof who ruined or is about to ruin their life and an opportunity to kill them all to advance is own scheme to get back at Providence and 47 is the hundred bullets, the tool for revenge that allows the Shadow Client's pawns to get away with murder. The Minutemen (Graves former team) is probably gonna be Providence's enforcers in new episodes when they decide it's safer to take out the ICA if Soders get exposed in the sixth episode.
Jordan Cross is an ex Providence agent.
Despite a strained relationship with his father, Jordan worked for Providence until he was sent to spy on Hannah. He fell in love with her, severed all ties with Providence and pursued a life of rock. After Jordan drunkenly murdered Hannah, Thomas bailed him out to keep Providence's existence a secret and possibly to recruit Jordan back into Providence. But Jordan doggedly refused to rejoin Providence and continued his rock career. That's why Thomas Cross wants the tape, he wants to blackmail Jordan into going back to Providence.
The Providence representative is "Mr. X" from Hitman 2.
He wears the same style of suit and his face and hair are pretty much the same (though it can be a little hard to tell due to the massive jump between 2000-era polygons and 2016-era photorealistic human render). It would also tie up a loose end from the previous games and finally give a concrete hint as to who that guy was.
Dr. Ortmeyer was a member of Providence.
The Providence official revealed at the end of the game that Providence knows a lot about 47's past, despite this information being so scarce that even the Franchise, who were influential enough to get one of their agents appointed Vice President of the USA, had a hard time getting their hands on Ortmeyer's research. In addition, it's heavily implied that the Shadow Client is an Ortmeyer creation, but he has a very personal grudge against Providence for some reason.
My theory: Dr. Ortmeyer was commissioned by Providence to create his genetically engineered super soldiers. He probably wasn't all that loyal to the organization, he just worked for them because they were willing to give him the money, resources, and secrecy he needed for his research. He was likely going to turn his creations on Providence at some point, had 47 not killed him first. The Shadow Client was also an Ortmeyer creation, and he seeks revenge on Providence for what Ortmeyer did to him and his brothers. Providence knows about 47's past because Ortmeyer kept them updated on the progress of his research to some extent. Once Ortmeyer was dead, Providence probably felt that recruiting 47 into their organization wouldn't be worth the risk (they don't want 47 to turn on them, too) and so they wrote the whole project off as a loss and left 47 to his own devices.
- Appears to be confirmed in the tie-in comic Birth of the Hitman. When 47 and 6 were captured at the farm as young men, the dialogue implies that Ort-Meyer is at least partnered with Providence, if not a an actual member.
Penelope Graves placed the hit on Viktor Novikov and Dalia Margolis.
It is unlikely that MI6 would neglect investigating the Shadow Client's anonymous tip or decide to assassinate Novikov and Margolis rather than capturing and interrogating them, as is the Shadow Client leaking out the intel to them knowing the risk of them doing so.
Perhaps Interpol reassigned Penelope Graves from the Rose investigation to work with MI6, where the Shadow Client reached out to her and recruited her as a mole. Graves, through manipulation, requests and snooping around, gained enough intel to allow Hall to hack into MI6 and steal the NOC list, which was then sold to IAGO. Then, impersonating an MI6 agent or official, she put the hit out on IAGO's heads before fleeing to the militia compound.
- Two background conversations disprove this theory. In Colorado, Graves ponders why the Shadow Client thought it was necessary to assassinate the IAGO leaders (and comes to the conclusion that it was merely a precaution due to Novikov seeing his face), while in Paris, Margolis has a conversation with the IAGO agent responsible for obtaining the NOC List (which she claims she got from listening to the commander she was assigned to talk in his sleep, after making him dream about work).
Penelope Graves and Carlton Smith knew each other back in Interpol
Whether as friends, rivals or even lovers.
Canonically, Agent Carlton Smith wasn't rescued in Situs Inversus
Erich Soders allied with the Shadow Client before Providence picked him up
Silvio Caruso was molested by his mother
After ascending to head of the family, Giles decided to use his wealth for the greater good by hiring out the ICA.