Doing what I want to do.

These days I make full use of my stamina into the ground.

Unclear, into the ground would be used like "I put a seed into the ground."

As I will have a Toeic Test again end of month, my time spent learning English has been increasing.

Toeic (Not a word in English)

On the other hand, my classes of The Open University of Japan are prone to delay.

If you take more than one class, classes. If you take one, 'my class at The Open University'

Of course, I sometimes hang out with my friends.

"Hang out" is like "To be in the company of my friends and do things together."

Play sounds childish unless you are playing a game.

Peculiarly, as I drunk a lot last night after the work with my colleagues, I was exhausted today.

Improper use of the word "peculiar."

I'm way too tired but I don't feel stress because I am doing what I want to do.

It might be happy I guess.

Try rewording this like "What I want to do will probably be happy." This is okay as it is.

Doing what I want to do

Titles don't end in a period (.).

These days I am running myself ragged. [Alternatives: "... I am working myself to death." "... I am stretching myself to my limits." "... I am running on empty."]

You can "run something into the ground", meaning to overuse it until it breaks down, but one doesn't use this expression with something abstract like "stamina."

I will have a TOEIC Test again at the end of this/the month, so I have been spending more time learning English.

When each letter of a name stands for a word (such a name is called an acronym), all of the letters must be capitalized. "TOEIC" stands for "Test Of English for International Communication."
Instead of "as" or "since" at the beginning of the sentence, it is more natural to introduce the second half of the sentence by "so".

On the other hand, my class of The Open University of Japan are prone to delay.

I'm not sure I understand what you're trying to say here. Could you say this in Japanese, please?

Of course, I sometimes play with my friends. [Alternative: "... sometimes spend time with my friends."]

The reason both スパイクさん and I are tempted to change this is that in English, "play" is used more to describe the activities of children than the activities of adults, except when speaking of playing a game (ゲームをする).

Peculiarly, as I drank a lot last night with my colleagues after the work, so I was exhausted today.

Both スパイクさん and I crossed out "peculiarly" because this seems like a very natural consequence of drinking a lot last night. "Peculiarly" would come before something unexpected.
The past tense of "drink" is "drank". "Drunk" is the past participle.

I'm very tired, but I don't feel stressed out because I am doing what I want to do.

The expression "way tired" is a slang expression used by some young people to mean "very tired", but I would advise against using such expressions until you have a better feel for the tone they produce. It is not standard English.

It might be happiness, I guess. [Alternative: I might be happy, I guess.]

"Happy" is an adjective; "happiness" is the corresponding noun.