How can I make this paragraph sound better.

There is not much wrong with this paragraph. Be careful of capitalizing common nouns. You use the words "many select few," which seems contradictory. Is it "many" or a "select few"? If you wish to emphasize that the number is extremely small, you could say "very select few."

An entrepreneur can be defined as an employer of productive labor. In Aravind Adiga's White Tiger, Balram tries to define himself as an entrepreneur in the terms of Indian society. It becomes apparent to Balram that his only way out of the caste system as a driver is by becoming an entrepreneur. All of his actions, motives, and justification center around his wish to be an entrepreneur, which means he would become one of the many select few of his caste who will become a man

Aravind Adiga's White Tiger Balram 

Comma missing? What is the subject of "tries"? I can't parse this.

tries to define himself as an entrepreneur in the terms of Indian Society. It becomes apparent to Balram that his only way out of the caste system as a driver is

to become an entrepreneur. 

"by becoming" is grammatically incorrect here, but sorry I can't explain why. 

by becoming an entrepreneur.

All of his action, motives, and justification center around Balram's wish to be an Entrepreneur << lower case, no caps>> <<insert comma>> which means he would become one of the many select few << "many select few" is an oxymoron>>of his caste who will become a man.

Perhaps:

In Aravind Adiga's "White Tiger,"  Balram tries to define himself as an entrepreneur in the terms of Indian Society. It becomes apparent to Balram that his only way out of the caste system as a driver is to become an entrepreneur. All of his action, motives, and justification center around Balram's wish to be an entrepreneur, which means he would become one of the select few of his caste who will become a man.

 "By becoming" is a preposition followed by a noun meaning "the process of coming to be something".

The "to be" verb "is" is typically (but not always) followed by another verb -- often an infinitive or a gerund verb form.

In "is by becoming", the "to be" verb is not followed by another verb.  This error often occurs because "becoming" seems like it should be the past participle of "become". ("Become" is the past participle of "become" and "becoming" is a noun.)

In this case (as demonstrated by Glorrierose), "is" should be followed by the infinitive "to become".