While it may be institution/field/country-specific, from the description I assume that you are applying to the program and do not have a prospective advisor who has already agreed to supervise you, nor a project onto which you will be allocated. In this case, that seems similar to the American "statement of purpose" (SOP).
The first guideline is: follow the guidelines. They tell you exactly what they want to see in your statement. You should absolutely touch on all of those points.
In fact, many statements add nothing to the profile exactly because the applicants fail to understand its purpose. The only way that you will have the reader's attention is if you talk research. The statement should not be the place where you state everything that they could have read (in a more appropriate format) in the CV. Your background is, of course, relevant, and you should absolutely include it, but only as evidence of your motivation and involvement with the field and the tools that you wish to learn or use during your PhD. The key to a good statement is to seamlessly merge your interests, background, experience, fit and the necessary tools to accomplish your goal. You should leave a strong impression that you have a solid (not necessarily deep) understanding of the field you are getting into.
So, for instance, don't write this (parentheses are my comments):
I have always liked to be boring and make people uncomfortable. Since
I was a child, my parents would roll their eyes every time they read
one of my essays. Rarely they would be able to finish. As I grey up, I
felt the need to contaminate my friends with my excitement, and made
them listen to incredibly long and meaningless talks. (Sorry to be
blunt, but nobody cares about your childhood.) I graduate in May from
Boring University and so far I have maintained a perfect GPA. In fact,
I took several courses in how to be boring, such as Being Boring I and
I have published a paper where we explored the mechanics of
self-expression in boring situations, along with Dr. Cheeky Jokes. (I
have your CV and your transcripts, I can see that.) Now, I want to
further specialize in making people annoyed, and Awkward University's
program in Social Engineering is certainly the best in the world (I
know, why do you think I work here?) and will give me a strong
background to make people uncomfortable while simultaneously being
boring (nice, but what do you bring to the table?).
Rather, maybe you should word your essay like this (note how I touch on all three topics while simultaneously referencing the background and fit):
The strong background that I acquired in boredom from Boring
University has motivated me to further specialize in awkwardness.
Working directly with Prof. Cheeky Jokes on the mechanics of
self-expression in boring situations, I noticed that awkwardness
emerges directly from personal interactions under these conditions.
Under the supervision of Boring University's Ethical Review Board
(ERB), we designed an empty room where we put people who knew each
other but hadn't talked in several years. Upon examining the data, we
found that they not only were extremely bored, but also engaged in
very awkward conversations about their past and how their kids had
grown up. This very exciting finding was later published in the
Journal of Awkward Situations. Now, during my PhD and motivated by my
previous research experience, I want to continue exploring the effect
of boring situations in the emergence of awkwardness. During my
research, I became interested in Prof. Clumsy's research on how to
maximize awkwardness in daily situations. In fact, I believe that
providing experimental validation for her recent work on finding the
optimal number of people who will maximize awkwardness in a highschool
reunion, which she modeled computationally, can be a major
breakthrough in the field. While Prof. Clumsy has already disclosed
that she is already working on that, I believe my experience with
experimental social engineering can be a solid addition to her group,
especially given that I have had thorough contact with ERBs.
Note, however, that in many cases the SOP is not heavily weighted in admissions, but that's probably field-dependent. And I am not in CS nor in the UK, so take my advice with a grain of salt.
