Let’s take things step by step. We ‘re talking about one of the powerful Shichibukai and if not the strongest, certainly one of the strongest swordsmen in the world of One Piece, Dracule Mihawk. We all understood his awesomeness, when in one of the first episodes he slashed and beat Zoro so easily, using just a little knife. 
He also trained him years later, in the two year gap of the time skip. That’s what we do know regarding him, but is there more to it? What is his role really? We haven’t seen enough, and that’s for sure. He certainly is a mysterious character.
His relation to the Straw Hats
It is quite obvious that in the times he met members of Luffy’s crew, his moods against them weren’t exactly hostile. He was once the dream of Zoro, the person he wanted to beat. In fact, Zoro lost to him and later became his disciple. Mihawk didn’t kill him when he could. Moreover, in the war of the best he attacked Luffy, but it seems it was just a way to test his strength, and not really an attempt to kill him. The pattern is that he either wants the Straw Hats alive, or at the very least is interested to see the potential they possess.
Blackbeard Pirates vs Red Hair Pirates
A battle that was basically foreshadowed at Marineford is the one between those two crews. Why is this related to Mihawk? There is a possible hint that Oda has given us when Shanks’s crew was revealed. In the war of the best we saw 10 members of the Red Hair Pirates and 11 of Blackberad’s crew. If it is intended for us to watch a one on one battle, it could be that one secret member is missing. Can that crew member be Dracule Mihawk? He has no crew, no pirate flag, no real ship even, yet he is considered a pirate by the World Government. There is a possibility he was part of a crew, and to be more exact the Red Hair pirates. He decided to leave them before Shanks was a Yonkou, in order to become a Shichibukai, an insider in the World Government working for his captain Shanks.
His relationship with Shanks
We do know that they are old acquiantances/rivals, he and Akagami.
Shanks knew where to find him when he wanted to contact him. Furthermore, the way he asked him to party with his crew, resembles the way a captain would go about treating a crew member of his. Also, when the Red Hair pirates arrived at Marineford he refused to fight anymore. This could as well explain why he hasn’t killed any Straw Hat, because in reality he needs them. On the other hand, a possibility exists that he was just a crew member once, and was actually disgusted by the way the Red Hairs do things, leaving them. That could explain why he didn’t want Shanks to contact him, neither party with the crew. Or maybe he was afraid his identity would be revealed to the World Goverment, should someone not trustworthy saw anything?