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.