How should I plan and structure a personal statement?

Personal statements are tricky: for someone who hasn't written one before, it's often difficult to work out what you should say, and how you should say it. Moreover, there's no 'one correct way' of writing them. Nevertheless, there are certain stylistic structures and formulas which you can apply when writing them, to aid you.
I believe that the primary purpose of a personal statement is to demonstrate a love for, and familiarity with your subject outside of the classroom. Everyone applying to university has sat A-Levels and GCSEs - you need to present yourself as someone who has engaged with your subject on a deeper level. For example, a student applying for English Literature courses will need to demonstrate an ability to engage with literary texts; meanwhile, a student applying for a philosophy course will need to demonstrate an enjoyment and understanding of certain philosophical issues. You don't need to seem like an expert - but you do need to seem like a nerd. 
How do you do this? Well, there are four main elements you can use to show off your - for lack of a better word - nerdiness. 
Firstly, you need to show a direct engagement with the subject itself. For an English student, this will involve discussing literary texts you have read outside the classroom, such as Hamlet or Pride and Prejudice. For a scientist or a mathematician? You need to show an understanding of your subject beyond your syllabus - discuss what interests your about your subject in particular. This is a chance to show that you enjoy your subject, and to show that you can discuss it in an academic manner
Secondly, you need to show that you have activley pursued an interest in the subject. This is where you discuss the clubs, societies, and activities that are particularly relevant to your subject. An English student may discuss their involvement in a literary magazine. A Philosophy student may discuss the debating society. A medic may discuss work experience or volunteer work. 
Thirdly, you need to discuss how the interests and experiences demonstrated in elements one and two have shaped you as a student and improve your ability to learn. Here you will discuss key talents you have, such as oranisational skills, leadership abilities, analytical thought, argumentative talents, interpersonal skills etc. 
Fourthly, you need to show an awareness of the overarching issues and abilities that are necessary to an understanding of your subject. For an English student, these might include the ability to understand the language of a text, the ability to engage with critics, the ability to compare texts with one another, and the ability to place texts in context. For a philosopher these might include the ability to understand and analyse other people's arguments, the ability to make your own arguments, and the ability to relate philosophical issues to the real world.
Now, how should these four elements be used? 
Generally, you should structure your personal statement so that each paragraph or issue you discuss contains one of these points. In one paragraph of my own personal statement, for example, I wrote: 
_{"Due to their enthralling nature, emotional tales are inherently persuasive and lend power to the arguments within; never is the cause of liberty better championed than in Animal Farm, where Orwell’s false neutrality and allegorical style persuade us to his view on totalitarianism. My study of Philosophy complements this aspect of my learning, and I plan to continue it alongside my study of literature, due to the insight it provides; the works of J.S Mill, for example, have led me to better understand Orwell’s 1984. Observing Winston’s tragic struggle for hope, I could not agree that dystopic fiction was “depressingly nihilistic” and went on to argue so in my Extended Project, relishing the opportunity for independent research; this experience has further inspired me, through sheer enjoyment, to take up literature at degree level."}
Note how in this particular paragraph, I use all four elements in concert. I begin by discussing the emotion power of literary texts, a general issue that is important to much of literary criticism (4). I then move on to discuss and analyse 1984 and Animal Farm, showing a direct engagement with specific texts (1). Here, I discuss Orwell's "allegorical style" in order to demonstrate my ability to analyse language (3/4 - it's a talent, but also an overarching issue). I then show how I have related my literary texts to philosophical texts, an activity undertaken outside the classroom (2/3 - it demonstrates the active pursuit of my subject (2) but also the ability to discuss textual arguments (3)). I then directly discuss my texts again (1) before moving on to discuss my use of them in my Extended Project (2), and finally mention how my EP has developed my abilites as a student. 
Note how I keep elements 1-4 in play throughout the paragraphy, including multiple instances of each (the relation to philosophical texts and the undertaking of an Extended Project are both instances of 2), and relating them to one another so as to ensure a discursive flow (I use my disagreement with the phrase "depressinly nihilisitc" (1) to springboard into my EP (2), before flowing smoothly into my personal development (2).
Each paragraph can be structured in this manner (with slight variations for introductions and conclusions), relating the four elements to demonstrate interest, engagement, undersanding, and ability. When structuring your personal statement as a whole, then, make sure to break your interests and experiences into categories one and two, creating a list for each. Then create a list of skills important to your subject for 3, and a list of overarching issues you consider important. Then organise a number of discussion topics, each of which should ideally contain one of each element (there's some leeway to use your own judgement, however). 
These topics will form the basis of your individual paragraphs. With this skeleton in mind, you can then progess to structuring each paragraph in detail. There are certain tricks to doing this - for example, 4 works VERY well as a means of providing each paragraph with an overarching theme, while 3 works as a very good means of closing off paragraphss. 
I can discuss these tricks in more detail during an online lesson, providing detailed stylistic support, and structural advice. I can also explain why - unless your instincts tell you otherwise - you should always, ALWAYS begin a discussion topic with element number one...