You could use your PS to explain these things - particularly if you didn't have a whole lot going for you in general. BUT (huge but) given your life story, there are way stronger things to focus on in your personal statement. You really want to talk yourself up in your PS. Big things in PS are your voice, outlook on life, and self-analysis/reflection on past events.
Don't get me wrong, those things (poor grades/lsat) are extremely important to mention in your application - but its better suited for an addendum than a PS. An addendum will give context to less appealing facts, ideally in a way (as you are attempting to do in your PS) that turns a weakness into a strength.
Personal Statement Outline:
Struggled early in life (family story) ~ Overcame it (later in life) [these two sections should be no longer than 2-3 para.] ~ Huge Achievements from my overcoming struggles turned into deep interests in... (law stuff) ~ Will succeed in law school and as a lawyer... Note: tie in, and reflect on, your family/culture throughout essay (which you are currently doing very well by the way)
Let me explain more about my early struggles that I touched upon in my PS, specifically as it relates to academia (tie in family story from PS) ~ How I overcame it ~ How this process turned my weakness into a strength ~ Why and how I will rock law school!
Those are my thoughts at least. I'd be curious what other think. I'd also heavily recommend talking to a pre-law advisor and/or writing center to get their perspective.
Good luck my friend. You are almost there!