I'm in the process of doing this right now. I've talked to several higher-ups at the KC up/down, and the STL TRACON. You can get hired off the street, all you have to have is a 4-year degree and be 21yo or older to be a candidate. However, they haven't opened it up to off-the-street for some time. My contact at the STL TRACON has been telling me for over a year there's been talk about opening it up, but it hasn't happened yet.
Having a background in aviation, especially a pilots license, will improve your ranking greatly, one accepted. I know two guys in the KC area that went to the ATC training in the same class, and got the same score on the test. One had no aviation experience and got only one offer. The other had PVT w/ INST, and had 33 offers.
Currently, I'm just continuing my pilot career route, but check on USA Jobs almost daily on the off chance they open it up to the public. Occasionally, they'll have it open to CTI grads, but even that's few-and-far between.