Roy Mustang and Riza Hawkeye's relationship can be traced back to long before the start of the series' main storyline. It goes back to before the start of the ever-so-important Ishval War, even. We don't know at which point in their lives they met for the first time, but we do know that Roy was an alchemy apprentice to Hawkeye's father in his younger days. He's the man responsible for his basic learning and the one who originally held the secrets about flame alchemy.

When Roy joined the Amestris army, his teacher expressed his disappointment in seeing his student becoming a dog of the military. However, I'd say it's obvious he still believed in Roy and his potential, for he was the one chosen one in case his flame alchemy was to be handed down. Hawkeye's father died in front of both his horrified daughter and Roy, the same day his pupil told him about his enlistment.

Roy: ... I see. If anything happens, you can visit the military authorities anytime. I'll probably be in the military for life.
Hawkeye: ... for life?
Roy: Yes.
Hawkeye: Please don't die.

Roy offers his support to Hawkeye, still a young girl and quite alone in the world. He'll go back to the military, but before that he shares his dream with her. A dream full of good intentions and naiveness, which makes Hawkeye decide to trust her back -- his father's alchemy secrets are carved in his skin, literally -- to that man who believes in protecting everyone with his hands. And thus Roy becomes the new Flame Alchemist and parts ways with Hawkeye (for these two are capable people, able to stand on their own when they have to), till some unlucky circumstances would make them cross paths once again.

     

Roy couldn't believe his eyes when he saw Hawkeye, the girl whose pupils shone hearing about his dream, his teacher's daughter and now a grown woman but very young still, standing in the battlefield with the eyes of a killer. The Ishval War made Roy rethink his life and set himself a new goal, changing a bit in the process. But Hawkeye became almost a totally different person during the war. After seeing what had become of the trust she relied on Roy (although not blaming him for it) and experiencing what she now had to do in the military's name, I believe she couldn't find herself anymore.

While Roy was taking the first steps towards his new objective, not even after the war was over could Hawkeye get rid of the guilt and the feeling of horror that remained within her. She then made a decision, in hopes it would push her forwards.

Hawkeye: [...] I have a favor to ask you, Mustang-san. Please burn and crush my back.
Roy: What are you... There's no way I can...
Hawkeye: At least!! If I can't atone, then at least so that it can't give birth to a new Flame Alchemist. So that the secrets on this back can't be used. And so I can lay down the bonds to my father and alchemy, to become Riza Hawkeye as an individual. Please.

The ties with alchemy she wanted to destroy, became a much deeper bond shared with Roy. He was the only person she could trust to hurt her back, and moreover, she wanted him to do it -- as much as it would hurt him too. I don't think Roy can ever say "no" to Hawkeye, and especially not to something as important for her as this.

So Roy, after learning his teacher's secrets straight from Hawkeye's flesh, he'd also be the one responsible for burning that knowledge, encarved on her skin -- an act of even more intimacy than the first one and with much more sentimental repercussion for both.