If I'm not too late, my advice would be to ask for his cell phone number/Skype/Whatever you use as soon as you can.
Even if he doesn't have any of the above, making a request like this shows that you like him and want to become better friends with him. By asking him for a means of communication, you're showing him that you're interested and like him as a person, which is important in forging a friendship.
The next time you're free and he doesn't seem busy, invite him out to do something fun, like a movie or trip to an arcade. If you enjoy his company, ask for his contact details so that you can plan a future event. If he likes you in return, which is pretty likely if he's having these nice conversations with you, then he'll give you these contact details and possibly return the favor by inviting you out in the future.
Good luck! 