You'll normally need good A-level grades (usually an A and 2 B's) in Maths and 2 Science subjects. (These are the minimum requirements. Many candidates will have 3 or even 4 A-grades).
This will get you onto a 5-year degree course in medicine. (It's possible to get onto the course without such a scientific background but you'll still need very good grades and the medical degree will take 6 years instead).
These 5 (or 6) years are then followed by a further 2-year Foundation Programme during which you can choose to start specialising in your chosen area of medicine (Psychiatry).
Following this there will be further training, typically lasting 5 or 6 years, specifically in Psychiatry.
A slightly shorter route into an associated field is available if you wish to consider training to become either a clinical psychologist or a counselling psychologist. For this you'll need three good A-levels to get onto a 3-year degree course in Psychology. This will be followed by a further 3-year period of training.
Links:
Psychiatry:
http://www.learndirect-advice.co.uk/helpwithyourcareer/j obprofiles/profiles/profile691/
Psychologist:
http://www.learndirect-advice.co.uk/helpwithyourcareer/j obprofiles/profiles/profile450/
Chris