To propose the idea of a relationship, the person should bring up personal feelings and what the other person means to them. The best way to breach this subject is on the basis of trust — if one person isn't sure about the other's motives or doesn't understand where the relationship stands, taking it further may not work out.
The person should talk in terms that make the situation about both people. If one only talks about oneself, it seems like they're coming from a place of personal gain. What they really want is to make the other person happy and a part of their life, so one needs to illustrate that. They can always show them that they care by buying them a favorite food or taking them to their favorite place before bringing up the conversation, and this will make them realize that the person has been listening to their wants and needs.
In the event that the other person doesn't want to join in a relationship or something goes wrong, a person needs to be gracious and understanding about it. The feelings may hurt, and it might be awkward to come back from, but lashing out or seeming ungrateful will make the motive for asking seem selfish and rude.