Your right to Voir Dire is drawn from the Texas Code of Criminal Procedure. Usually, your lawyer asks questions of the jury and then challenges the prospective juror for cause. Once your lawyer finishes questioning the jurors and making his/her challenges for cause you and your lawyer can use 10 peremptory challenges. Usually, courts do not appoint Forensic Psychologists to assist you with Voir Dire, although you may be able to hire one on your own dime. Additionally, jurors usually fill out a questionnaire that asks them basic questions about their background. Finally, your right to Voir Dire a jury is not limited by your choice of lawyer, but the judge can limit it according to his/her discretion. Good luck.
There is not a full investigation of jurors done in any case, including capital murder cases. The State will likely run a background check and your lawyer can ask that the judge order the State to share that information. That will provide them with criminal cases with which the person may have been charged, but nothing more. To determine about their life experiences, the lawyers question the potential jurors.
You will not have a forensic psychologist or any other type of expert to help during jury selection unless you hire one. This is not considered to be an area of expertise to which you are entitled during jury selection.