Assuming that your brother's total income (from both forms of employment) won't push him into a higher tax bracket (which would seem to be very unlikely, based upon what you've written) he'll be taxed at a rate of 20% on everything he earns from his second job. (He won't have any tax-free allowance as that will all have been swallowed up by his first job). So he needs to set that amount aside in order to pay his additional tax bill (on top of what he's paid via PAYE on his main job) at the end of each financial year.
He needs to start by getting a Government Gateway account but, rather oddly, he can't actually register as self-employed until he's started being so. Then he should register straight away. (Getting a Government Gateway account is annoyingly tedious but actually quite straightforward. Your brother needs to ensure that he's recorded the log-in details somewhere that he won't lose them, otherwise he'll end up in a wild panic when the deadline for submitting his tax return approaches. Written from experience!)
At the end of each financial year he'll need to complete a tax return. He should definitely opt to do it online as it's far simpler than the paper version. (With the paper version you have to work through loads of questions just to find out whether they apply to you or not. The online version asks a few simple questions to start with and then only show you any further questions that actually apply to you). He'll need to fill in the details which are on the P60 that his employer should give him at the end of each year, so it's very important that he doesn't forget where he's put it. (Also written from experience!).
Your brother will probably also have to pay a small amount of self-employed National Insurance on his second job but, as it's nowhere near as much as is paid on an employed job, it's nothing much to worry about.