Once the war is over, Roy gets assigned to East City and begins gathering his little circle of trusted comrades who would help him reach his goal. Among those comrades is Riza Hawkeye, of course, and the only one we actually get to see Roy asking. It pains Roy to see how much Hawkeye has changed, for the worse and due to things he in part had contributed to. She's hurt and bitter, and willing to lie to herself to keep going.

Roy: I'm thinking of recommending you as my aide. I want you to protect my back. Do you understand? To entrust my back to you means that you can shoot me from behind anytime. If I step off of the path, shoot and kill me with those hands. You are qualified to do that. Will you follow me?
Hawkeye: Understood. If that is your wish, then even into hell.

Roy wanted to give her hope again, a new dream she could trust and dedicate her life to, without having to resign herself to a life of self-sacrifice serving the military with no hopes of making it any different. He gave her the choice to aim for something better, not just for everyone, but for herself too. She needed it in order to save herself.

Not only that, Roy also entrusted his back to her -- just as she had done with him a long ago. By trusting her back to him back then, she relied on him her father's secrets and basically the wish that that power could be used for the greater good. Roy failed on his first try, though. By returning the gesture to her and putting his life and death in her hands, Roy is trying to gain back Hawkeye's trust. Trying to prove to her again that changing things is possible and that they can do so if they work together. He also wants to atone for whatever pain his past failure inflicted upon her.

     

During the time between Roy's assignment to East City and the beginning of the plot, we can see a lot of changes in Hawkeye again. Even way before that, when both visit Resembool and find the Elrics, Hawkeye's progression is already quite visible. She became increasingly cold during the War, till the point when she seemed almost expressionless and empty, having been deprived of dreams and disappointed by the person she trusted most.

However, those years in East City act as rehabilitation for her. We ignore most of what happened there; but Hawkeye, although still serious and strict in the outside, is warmer, comfortable about what surrouds her and relaxed. Being near Roy, her new comrades (a lot more easy-going and without the weight that is attending a war in their hearts) and also observing and being part of a new dream in the making, transforms her into the Hawkeye we know now.

Hawkeye: The military isn't going to take them away. It will be their choice whether to come or not. To tell the truth I don't like the military either. Because at times, I am forced to take lives.
Winry: Then why are you in the military?
Hawkeye: There is someone I need to protect. It was not because I was forced by anyone, but it was my own free will. It is my own choice to pull the trigger for the person who I must protect. Until the day that person accomplishes his goal... I will pull the trigger without doubt.

This is a pretty good example for that. Hawkeye's able to discuss calmly with Winry, reassuring her that Roy doesn't mean any harm and that it's a choice what he's giving to the brothers, the possibility to stand up and recover. She knows this because he also offered her a way out, and it's very important for her to let Winry know that it was her choice, just as it'll be Ed and Al's choice.

Also, Roy needs to know he's not pushing anyone, that it's always other people's free will speaking, and not him forcing the ones around him. Hawkeye reassures him from time to time, he needs to know she's there because she wants to and not because he needs her to be.