You need to register as a sole trader with HMRC
https://www.gov.uk/set-up-sole-trader
which basically just means registering for self-assessment
https://www.gov.uk/log-in-file-self-assessment-tax-return/register-if-youre-self-employed
(Click 'Register online')
You'll be asked to give a name for your business but you don't need to think up anything original; you can simply use your own name as your business name.
Then you simply keep records of everything you've earned and the expenses you've incurred and fill in a tax return online once per year. (It's amazingly simple; the whole process only takes a few minutes).