I went straight from Uni to a handling agent - starting as a controller. From there, promotion - bit of company switching, before getting my job now as a Senior Ops Controller.
An interest in aviation, good memory, like working horrid hours, contientious, switched on cookie, you know that sort of thing! Bear in mind at the end of the day although it is slightly different it is still a desk job. If flying's what you really, really want to do, pull out all the stops and go for it.
Ops in an airline probably isn't the place to get hours in, a flying school is.
Good luck.