I would just ask them what their schedule is for new hires, ideally you would have asked that in your first interview(s), do you have any contract, what does it say ? It's also easier to find a job while you have a job, currently interning at X company sounds better than unemployed these past X months, and you can use any offers you get from either side as leverage, so I think the when is now, the how is somehow more delicate, but you need to know when they will hire you and how much will they pay you ( whats the starting salary for a designer) . I would write the boss/hr department a nice letter with your questions and maybe forewarn them you are sending it while at the same time pursuing other opportunities. 3 months is a somehow average internship length so your request shouldn't be seen as coming out of the blue, especially if they are not paying you, hopefully others will chime in.
Best,
-K