Is this sentence correct?

Grammatically, it's OK, if a little loose.

No, its a run-on sentence that should be made into two sentences. Thus the inclusion or exclusion of the word "that" makes no difference.

(that) I'm ambitious someone
who always dream big
vb chase=pursue till last breath
to make dreams come true.

Yes

Yes, you can eliminate the word "that."
Your sentence is wordy. Keep it to the point without adding unnecessary information.

You can say I'm too ambitious, but I dream big, and I'm willing to chase my dreams until my last breath.

The omission of "that" in that context is fairly common in everyday speech, but might be regarded as "informal" in some written contexts.

I would finish: "....chases his dreams to his last breath, to try to make them come true". But your form is not "wrong".

It is incorrect and it is not one complete sentence.

This is how I would write:

Being highly ambitious, I always dream big and will chase my dreams till the last breath to make them come true.

no

No.
You need to simplify, because you have too many phrases for this sentence to be clear, and because you are modifying "breath" (singular) with "them" (plural) and "come true"... you wand DREAMS to come true, not your "last breath"...
The last phrase is too much. You can say "... chases his dreams until his last breath" or "... chas"es his dreams to make them come true
Also, "man" or "woman" makes a stronger sentence than "someone", and I would drop the word "too", because there is nothing in the sentence that supports the use of the word -- nothing that indicates you are TOO ambitious, rather than just ambitious.
"Chasing" a dream doesn't indicate someone is ambitious, like the word "work" does. "Chasing" usually indicates someone who lacks focus. So, I would change the word "chases" also.

You can say I'm an ambitious woman who always dreams big, and who works to make her dreams come true.

A comma after breath is appropriate, but it's what is called a run-on sentence. IT's too long, so try breaking it up into two sentences. End sentence #1 with the word "dreams," and then finalize your thought in sentence #2.