How to structure the personal statement?
Your personal statement or statement of purpose has a huge impact on your chances of securing a place at the university of your choice. The best way to structure it is to break it down as follows:
1) The Hook - this is the all-important first paragraph that grabs the admission committee members’ attention and makes you stand out from the rest. Don't be afraid to get creative with your writing here, you need to construct a narrative that stands out.
2) Are you telling a story? Admissions officers are human too! Don't relay your life story in a dull rote fashion.You need to communicate how you have got to where you are today and where you are going and why!
3) Support your story with data and facts - back up your narrative with actual data to support your claims. be careful not to go overboard and make the statement a list of facts and figures only.
4) Avoid the use of clichés - pretty straight forward, instead of a lame clichés, give an example of something you have actually done and back it up with hard facts.
5) Tailor your admission essay to the university you are applying for - every college is different and every course is different so make sure you do not just talk about how great you are, but what is great about the institution you are applying for.
6) Appropriate language and tone - You are not drafting the constitution or bill of rights, strike a balance between making sure your application is taken seriously and communicating a little of your personality.
7) Be Truthful - avoid the temptation to completely make things up.
8) Are you convincing - ask yourself if this is really the course you want to study and the place you want to study it? If you cannot convince yourself, then you will not stand much chance with an admissions committee.
9) Is your statement logical and coherent? The statement should tell you personal story. There is no set format, but be methodical as you progress though your key points, building a portrait of yourself.
10) Is the language perfect? Speak in plain English and always proofread your finished piece. Better yet, have a professional proofreading service give your statement the once over.
This answer is based on an awesome guide to writing a killer Statement of Purpose from Vappingo.com. There is also a free handy checklist to download.
Edit: I misread the question and interpreted it as being about including personal aspects in the usual statement of purpose, rather than writing a separate personal statement. Berkeley is unusual among universities in having standardized university-wide on two essays for graduate admissions, a traditional statement of purpose and also a personal statement, which often deals with diversity and inclusion issues. The advice I give below is aimed at other universities with a single statement of purpose, while Berkeley may be looking for something different in a personal statement.
Don't worry about including all these things in your statement. You don't need them all, and trying to cram them all in will very likely lead to a worse statement. The goal is to write something coherent and well organized, rather than a brain dump of everything that might be relevant.
Overcoming barriers is worth mentioning, but it should not be the focus of your statement. The purpose is just to put your accomplishments in the context of the opportunities available to you, so the admissions committee can judge your potential even if you haven't had the same opportunities as other applicants.
Academic service regarding equitable access could also be worth mentioning, but in most cases it will be at best a very minor factor. It could in principle end up as a tie breaker between two candidates, but it's not going to overcome any weaknesses in your application. Mention it if you have something impressive to say, but don't give it a disproportionate amount of space.
Understanding the barriers faced by others is generally not relevant. It's a commendable personal trait, but not one of the admissions criteria for graduate school.
Evidence of your research focusing on underserved populations or related issues of inequality
This is in an entirely different category from the other topics you mention. The motivation for and impact (potential or past) of your research are of real importance, and you certainly need to make this clear in your statement of purpose if you have something noteworthy to say.
And more importantly, how your personal background and experiences inform your decision to pursue a graduate degree.
If you are talking about telling autobiographical stories to explain how you arrived at this point, it's almost never a good idea. Discussing your academic experiences in college is worthwhile. For example, you might discuss what you learned from different projects (summer research, senior thesis, etc.) and how your plans have taken that into account. Talking about high school is generally a bad idea: it looks bad if after college the most relevant things you can think of to discuss are from high school. Don't even bother mentioning anything earlier. Nobody cares at all who inspired you as a child or what your childhood dreams or accomplishments were.
And is there any different between this statement and the SOP for undergraduate applications?
Yes, it's almost completely different, which means advice for undergraduate application essays is at best useless and at worst actively harmful for graduate applications. At least in the U.S., undergraduate admission often takes into account what graduate admissions committees would consider to be ridiculous fluff (e.g., Caltech asks applicants about "an unusual way in which you have fun"). Your statement of purpose for graduate school should ideally not contain any fluff, and if it includes more than a tiny amount then it will hurt your chances.
You mentioned in a comment to an older question that you were planning to apply at UC Berkeley. Hence, @AnonymousMathematician probably links to the particular situation your questions results from; although at schools where you have a like split between personal statement and SOP this might be similar.
It seems that this part of the application is meant to put a deserving candidate who faced obstacles (gender, race, financial, ...) but persevered on equal footing with those who didn't face such difficulties. In addition to influencing whether or not you might be admitted, the description mentions that your statement will be considered when fellowships are allocated.
It's a common concern in the U.S., but might not be something you, as a foreigner, are used to. In essence, though, in your country people are dealing with comparable challenges (an Indian friend who, despite being poor, found himself at a top IT and surrounded by very wealthy classmates certainly did, say). If you are part of any minority or group facing occasional hostility in your country, if you are gay in a country where this is still a problem, if you were a woman in a country still placing bounds on women's progress, if you had to work part-time to support yourself (or were supported by poor parents and felt guilty about it), etc. - this belongs into your personal statement; and if you assumed a leadership role, for yourself or others, in such matters, this would be particularly useful.
Now, none of this might apply. Note though that the description on the web page linked mentions that "you can repeat information about experiences in your research statement." So in this case do that, and add a bit about your background (e.g., parents' job or a mentor influenced your career choice; when taking class X, I realized that this is what I liked). In your particular case, you mention in another old question that you participated at a math Olympics, while being unnecessarily humble about not winning a medal. That strikes me as a good story to tell. I'd focus on the related travel and time in a foreign country (different food, habits, ....), and how meeting all those attending from over the world was fascinating. If you made a friend from another country at the Olympics, good story too.