Obviously this is the million dollar question, but as a general rule I would seek to follow these 3 steps after you have finished writing your essays, to ensure they are as clear and persuasive as possible:
Firstly: Get in the habit of reading your essay with different eyes.
In my opinion, there are two ways of reading. The first is what we do when we see an advert, or subtitles in a film: we read passively, without concentrating or processing. Though it may seem the opposite, this is actually highly useful for proof-reading your essay. You quickly go through your text, scanning it for any glaring errors in syntax, punctuation, spelling or argumentation. Let them glare at you until you get rid of them. Also, circle any paragraphs or sentences that would take you longer than 5 seconds to amend, then move on – you will come back to them in step 2. This has the benefit of saving your high focus exclusively for things that require it.
Secondly: Focused reading – but only where absolutely necessary.
The second type of reading – registering and analysing every word – should be saved only for any areas you have circled and crucially for the introduction and conclusion as well. Focused reading demands a lot of effort if you do it properly, and you should take time to ensure that all arguments are clear, fully nuanced, concise and answering the question in the best way possible. This is what ensures the top marks. If you are criticised by markers for failing in these things, it is simple: it means you either circled the wrong things whilst scanning in step 1, or didn’t take enough time on step I also cannot stress enough how important the structure and clarity of your introduction and conclusion is, not least because more often than not this is the only part of your essay that teachers/examiners actually read! (Going into detail on how to do this, however, would require far more space to explain!)
Thirdly: Do some final checks.
Once you have completed steps 1 and 2, I always repeat step 1 to ensure all my changes have been done without silly mistakes, before re-reading a last time with the freshest eyes possible (a 10-minute break usually helps). Here I double-check that all arguments are clear and respond to the question, that my introduction is engaging and my conclusion powerful.