The short answer is, if you have a felony conviction then you are not eligible to have a concealed handgun license in Texas. If you are able to get your felony pardoned and expunged then you may apply based on the fact that you technically no longer have a felony on your record. But if you have a felony conviction, you are not eligible.
Here is link on pardons and paroles that may be of interest to you. http://www.tdcj.state.tx.us/bpp/exec_clem/exec_clem.html#WHAT_IS_EXECUTIVE_CLEMENCY
Below is a section of the applicable law on eligibility for Concealed Handgun License.
Texas Government Code
Sec. 411.172. ELIGIBILITY. (a) A person is eligible for a license to carry a concealed handgun if the person:
(1) is a legal resident of this state for the six-month period preceding the date of application under this subchapter or is otherwise eligible for a license under Section 411.173(a);
(2) is at least 21 years of age;
(3) has not been convicted of a felony;
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