Not done it myself but know someone who has and who was successful at admin assistant level (she had no degree). Basic process is application form - long and complicated so give yourself plenty of time to fill it in. If you get to the next stage (bearing in mind they have hundreds of applicants for every post), you will be asked to take a test - you'll have to go to a venue to sit this (its basically a series of aptitude tests) - if you get through that stage you will then be called for an interview which I imagine is basically like any other job interview - if you get past that there may be a second interview although this probably depends on what level you are applying for. If you are ultimately successful, before a formal offer of employement they will carry out detailed security checks on you - police interviewing your neighbours etc.
I know that all sounds horrendous but you should definitely go for it. Loads of people apply so the odds of you being successul may not seem good but bear in mind they have to appoint someone and you could well be that someone.
The whole process is likely to take at least 6 months so I'd suggest applying for lots of other things too and in that way get some interview practice in before you're called for an interview with MI5. It will probably take 3-4 months before you get called for an interview.
Anyway, go for it and good luck. :-)