My biggest suggestion is that this letter should be much shorter. Your target audience is administrators. They have to make these decisions, but it is just a job to them. Yours is not the first of these letters they have read. Do them a favor and make your case succinctly.
You want to tell them that you desire to continue at the University, that you take responsibility, and that you have a desire and a plan to improve. You have hit all of these important points. There is a lot you can take out without taking away from any of these points.
For example I would say "I take full responsibility for my actions... " (without the rest of the sentence) and take out sentences like "Once becoming a student at the XXX, my trials began to increase."
The third paragraph is saying that you had some difficult life circumstances. The same point can be made with much fewer words (the details aren't important to your target audience).
The fourth paragraph is pretty good. Get rid of the phrases "Because I was having trouble mastering the material..." and "unfortunately, I did not reach my goal" (they are not needed and add nothing to the letter).
I would greatly shorten the last paragraph. The idea is good, you want to convince them that you are making the necessary changes. There are just too many words. Make the point with a couple of steps you will take, and leave out so many details. If they take you up on your offer, you will discuss the details then.
Believe me, the shorter the better.