In regards to developing relationships between the characters, I would say that it is about 30% up to the players and 70% up to the DM. When I DM, I take a character's backstory and see if there's anything I can do with my campaign to draw them in.
For instance, I have a Cleric who was orphaned at a young age. Cool. He could meet his parent(s) way later on in the story, with lots of foreshadowing. Other players will catch these clues and relate that Cleric has a dark past that isn't quite history.
It would then be up to the player to PLAY that role. An orphan is typically broodish (Luke Skywalker, Harry Potter, Batman, Cinderella, etc) or has a good reason not to be and will cling all co-dependently to that. With both the player and the world around him giving the feel that this character DOES exist, the other players will have now have a reason to temporarily suspend their disbelief. If EVERYONE is doing this, it can lead up to a really great roleplaying campaign.
As far as developing relationships out of game, well... use the same social skills you would use in real life. It's not all about D&D, so try to discover the other player's interests and discuss them... or do them! If they like blasting people in Planetside 2 if they're not looting treasure, DL the game and play with them!
I hope you've found this useful.