Bioinformatics is a popular field at the moment, there are loads of departments around the UK offering Masters and PhDs (see above posts).
I think your background would make is possible for you to go directly to a PhD, especially as your department will have bioinformatic courses that will bring you up to speed. HOWEVER, you will be competing with people who have the advantage with their dedicated undergrad degrees and/or projects.
I would go for a Masters or a 4-year course if I was you!