This anI’m doing this exercises because I'd like to become better withimprove my English's skills.

It's necessary to domake mistakes and fix quickly and this exercise ofthem. This writing thatexercise obligates me to think oin eEnglish and fouind words that express the ideas that you areI am trying to write.

Actually, while iI'm writing this, I feel good, quiet, and relax,ed. I just I need to find time to do this.

Currently, the place where I'm working gives us a license byif we practice english, so the idea is to take advantage of that and obviously keep writing in this page.

I’m not sure what you meant by “license”.

This anI'm doing this exercises because I'd like to become better with my English's skills.

"English" works as an adjective by itself, so it doesn't need a possessive apostrophe to modify a noun.

The same is true of most other language words.

It's necessary to domake mistakes and fix them quickly, and this exercise of writing that obligaterequires you to think oin eEnglish and fouind words that express the ideas that you are trying to write.

"make mistakes" is more natural than "do mistakes"

Actually, while iI'm writing this I feel good, quiet, relax,ed, I just I need to find time to do this.

"relax" is a verb, "relaxed" is the adjective.

The word "I" is always capitalised, even inside contractions like "I'm"

"just" here is an adverb for "need" so it goes after the subject ("I") and before the verb ("need")

Currently, the place where I'm working give us a license by(certificate?) if we practice eEnglish, so themy idea is to take advantage of that and obviously keep writing in this page.

A license is normally associated with permission to do something. I'm not fully sure what job you work as, but certificate might be a better word.

My sSecond tTime

Titles have special capitalization rules

This anI'm doing this exercises because I'd like to become better with my English's skills.

It's necessary to domake mistakes and fixcorrect them quickly, and this exercise of writing that obligatforces me to think oin eEnglish and fouind words that express the ideas that you are tryingI want to write.

We fix things that are broken => I need to fix my broken window
We make mistakes, and mistakes are corrected.

Actually, while is I'm writing this, I feel good, quiet, and relax,ed. I just I need to find the time to do this.

"I" is always capitalized.
Describing this present moment => currently
"Actually" is used for something that may be a little unexpected. Some examples:
------Mark said he saw a duck at the lake, but it was actually a swan.
------The weather forecast said it was going to rain, but it was actually sunny all day.
Since there is no reason for your reader to assume you would not be feeling good, no need for "actually."

You also don't need "currently" here, because "As I'm xxx" already means you are describing this point in time.

Currently, the place where I'm working give us a license byn incentive to practice eEnglish, so the idea is to take advantage of that and obviously keep writing in this pageposts.

English is always capitalized.
I'm not exactly sure what you mean by "a license" here, but a license is like a permit that proves you have the right to do something. For example, a driver's license is required to drive, a commercial license proves you have a right to sell merchandise, etc.
I substituted "incentive" which means your work is encouraging you to do this and presumably will reward you somehow.
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