Starting from the top I'm just going to point out that I prefer the Arbiter to the Masterchief as a protagonist in the Halo universe but now that that's out of the way:
I would portray the Chief pretty much as he was in the books, quick thinking, determined and very quiet. One major thing to his character and indeed all the Spartans is that he was abducted as a child and sent off to undergo a lifetime of training from hell while his parents believed him to be dead. With no childhood to speak of pretty soon the only aspect of his personality remaining would be the new enforced militaization.
His major flaw would be obvious, the inability to socially interact with other people, his whole life and everything about him is now exclusivley about the military and fighting, hence part of him would have to be hoping to die in the war because if he survived the end of the war he can't exactly just enroll in a course at university and find a new job. He's a Spartan, created solely for war, bred to die in combat.
This would explain why he shows so little personality in the games, it seems unlikely that he would have a favourite book, or Tv Show, or happy memories of childhood. All that really remains is his training and hence it kind of makes sense that he would develop a strong emotional bond to Cortana, the AI construct that was assigned to work with him exclusivley. The moments where he talks with her are rare moments of his humanity long lost after being made into a weapon.
And yet despite all this, does he whine? No, he fights, it was what he was made for and he is determined to fight for his sense of duty.
The games don't do a great job bringing that across but really that's who Masterchief is.