I'm sorry, but no--you cannot be licensed as an attorney if you are a convicted felon, regardless of how long ago your conviction might have been incurred.
Possibly. A felony conviction is not an absolute bar to becoming an attorney.
The question in front of the Texas Board of Bar Examiners is whether an applicant possesses the current good moral character to be an attorney in Texas. (http://www.ble.state.tx.us/Rules/NewRules/ruleiv.htm)
After five years have elapsed from the end of probation or completion of the sentence, a person with a felony may apply to attend a Texas law school (or, if a graduate of an ABA accredited law school in another state, to write the Bar Exam). The Board will then investigate and make a determination based on many factors, including the type and severity of the offense, and the life of the application since the offense.
It is not easy, but it can and has been done. You will almost undoubtedly need the assistance of a Texas attorney who specializes in these matters (which I do not, although I know attorneys who do).
If it is your dream to be an attorney, don't give up. Know, however, that you will have a long, hard journey in front of you.