The opportunity for success was so minuscule; and it became increasingly obvious along the way.
I think you should use a comma here instead of a semi-colon. It works as one sentence. If you want to add a longer pause, use a dash
-- (I hope that's what they're called).
The victims of this plight were scattered helplessly on the periphery for all future hopefuls to witness. However, my situation was plenty worse than most of those who came before me.
I think you might be too vague here so far. Someone reading this might get lost in these two sentences and forget them as they move along to the rest of your essay.
The likelihood of a child, that was abandoned before it could even defend itself, evolving into a confident and respectable individual was nearly impossible.
"The likelihood that a child, abandoned before it could even defend itself, would evolve into a..." <- I think this might sound better.
My lack of parents, siblings, and family altogether, further exacerbated the situation.
You already mentioned this in the previous sentence. You might want to be a little more blunt and specify that after you were abandoned, no one came to pick you up. I think, in this case, directness is what fits.
I quickly made peace with my dire circumstances and vowed to strive for excellence because mediocrity was unacceptable.
Maybe you should add a transition word like "however" here.
I was oblivious to the notion that things would only become progressively challenging.
You might not even have to include this sentence.. you've already made this clear.
I think you should also tell us what Jesse was to you: foster parent? Before you elaborate about him.
I settled for a private university out of high school that allowed me to escape the blinding cold that once distorted my vision and blurred the road ahead.
Did you really settle? Or did you enjoy this change?
And what do you mean by blinding cold?
Overall, I think you have a story that has potential. It might be better though, if you emphasize a bit more on how this whole situation made you feel. You've told us a lot about what happened and how you reacted... but tell us more about the conflict that went on within you.
Hope I helped :]