It is possible to write a winning essay without having read any writing guides and without getting any help from others. But this is inadvisable.
1. At the very least, get the reactions of one student or peer reader and one reader above that level (a teacher, parent, boss, or professor). Best of all, try to find a reader who is most like the readers who will judge your essay. Try to engage them in conversation about the essay's strengths and weaknesses.
2. At the very least, read your essay out loud to catch typos and, more importantly, to hear the tone and flow of the essay. Try to read it in the presence of a peer and/or a superior, and have that person read it back out loud to you. Remember that the reader of your essay will read as the essay looks on the page, not the way you imagine it sounding in your head. Even so, note that one of the judges might read individual lines or sentences out loud in order to persuade other judges that your essay deserves to win (or not to win). Make sure that your worst three or four sentences are still readable. Just as important, make sure that your best three or four sentences are memorable and of prize-winning quality.
3. At the very least, put the essay aside for a minimum of 24 hours after it is "finished," don't even think about it, and then return to the essay with a renewed mind and a fresh eye. Give yourself 48 hours or more if you can. To make the most of that time, give others the essay so that they can suggest some editing improvements for you to consider. GradeSaver has extensive experience providing such advice; let GradeSaver give you specific directions for editing your essay to the level required to win.