In the last five years, I have reviewed more than twenty personal statements that high school students wrote for admission to UK universities. There were many overlapping issues that I thought would be worth aggregating. A small caveat is that I reviewed engineering and natural sciences statements, so take my tips with a grain of salt if you are applying for social sciences.
Focus on academics. Degrees in the UK are highly specialized, so you have to demonstrate that you can dive deep into your subject of interest. During my time at Imperial College, I barely managed to take two non-CS courses because I was overwhelmed with a high CS workload. This education model is not for everyone, but if it is for you, make it clear in your statement. I would suggest focusing 80% of your statement on why you want to pursue your subject. Off-topic extra-curriculars should be treated as a nice addition, but should not be the core of your text.
Demonstrate passion through action. Do not say that you like your subject, demonstrate it through what you have done. Did you prove a nice theorem for a competition? Great, write about it. Did you write a compiler for a toy language? Awesome, include it. Did you make a robot follow a line? Sweet, put it in. You get the story, the more you can demonstrate your passion via projects, reading, competitions, etc., the better. Mention some extensions of these projects that the university will allow you to finally learn about.
Polish your English writing. Chances are that your native language’s writing style leaks into your English texts. This problem goes beyond using the right grammar and vocabulary. Even if you are a native English speaker, make sure that your statement loosely follows an academic English writing style. Academics (i.e., the people that will read your statements) enjoy reading concise, clear, and structured texts. There are many resources on this topic online, but here are a few links that seem useful: topic sentences, structuring paragraphs, academic writing.
Write a coherent story. People tend to write paragraphs that do not link well together. Each paragraph would be a self-contained story with an introduction and a conclusion, making the personal statement similar to an extended CV. Good statements take people on an intellectual journey which makes sense both on a paragraph-level and as a whole. Try to find some thread that you can use to link your paragraphs and make them flow.
Avoid clichés. While writing your statement you should ask yourself the question: Can anyone else write these exact sentences? If the answer is yes, you are probably doing it wrong. This is an exaggeration, but the more unique (while not too weird) your text is, the better you stand out among the thousands of applicants that are competing for a place.
Strong opening and closing. The admissions officer that reads your statement will likely have read a few hundred others before reading yours. It is important to have memorable opening and concluding paragraphs to “wake” them up. Coming up with these paragraphs often introduces a writer’s block, so do not spend too much energy on them. Quickly write down a few ideas and iterate as you write the rest of the personal statement. The right ideas will come to you as days pass.
Why did you choose the university? Being specific about why you like certain universities goes a long way. Everyone applies to the University of Cambridge because it is a top institution, which means that it is a bad reason to write about in your statement. Are there particular courses that you like? Is there research that you would like to explore? Is the department you are applying to particularly good at something? Specificity shows that you are mature enough to consider reasons beyond rankings.
I hope these tips help. You should aslo check out the following resources for writing personal statements: UCAS tips, BridgeU, British council. Good luck with your application!