I'm assuming that you already hold an engineering degree. That'd be step one. After that, it's like getting any job that you may want: make sure you have the minimum qualifications and build a professional network.
So what are the minimums? It varies widely depending on the organization, but if you have a degree in aerospace engineering and a Private Pilot license (or at least a good understanding of what pilots do), you're well on the way. MEs or EEs also have a pretty good shot depending on the program. Lab or wind tunnel work will also help. Then it's just a matter of getting as close as you can to the flight test organization. Any position where you analyze flight test data or provide engineering support for the maintenance folks will put you in the right circles.
As for the medical, here in the states, most companies require a 3rd class if you're going to fly on the test aircraft (the FTEs are primarily in the telemetry room for a lot of large programs these days). I'm not sure what the JAA equivelant entails, but for the FAA, a 3rd class is just a routine checkup and a detailed history. As long as you're in reasonable health, have normal BP, and aren't colorblind, you usually don't have a problem.
Finally, I'd highly recommend talking to any FTEs or test pilots you can find to make sure this is something you want to do. Everyone I talk to says "oh, you must have a really exciting job." But if things get exciting, it means we haven't done our job correctly. And you should also be prepared for long hours, odd hours, and high pressure. 6 days a week is the norm at most places I've worked, and 7 isn't unheard of. If you're flying missions that require still air, you've gotta have wheels in the well at sun up. Depending on how long the preflight checklist is, you could be looking at 3AM briefs. Or you could be called upon to work nightshift to ensure that the aircraft is correctly configured for test the following day. And keep in mind that you'll be the last thing in the way of delivering an aircraft that is probably many months behind schedule before it even came to you. Program managers, VPs, and in some instances CEOs will know you by name and be asking questions when things don't go according to plan (and they won't).
But if you can handle all that, flight test is one of the most interesting jobs that you can have as an engineer. Sure beats driving a CAD scope all day. Good luck and let us know how it goes.
-Shannon