I agree that your research should begin with your examination of other pro-death penalty critics. Once you have gathered this research, you can choose which critic you feel the most connection with. Build your research around that idea.
Since the avowed purpose of the Death Penalty (as stated by those who support it) is that it serves as a deterrent, your thesis supporting it should reflect this fact. Your research should cover particularly heinous crimes for which the penalty was imposed, such as the Oklahoma City bombing, child murders, etc. To the extent you can do so, your thesis should negate the argument that it is based on revenge rather than deterrence. You should also be prepared to argue that its deterrent value far outweighs the cost of its imposition. Finally, be careful to avoid hyperbole; keep your argument matter of fact and free from emotional appeal. This may be hard to do, as there are few issues more emotional than that of capital punishment; still frame your argument in terms of compelling reason as opposed to impassioned pleading. Good luck!
The first thing you will need to do is research several arguments in favor of the death penalty. Then you can use them to combine together to create one comprehensive argument. For example, let’s say you decide to focus on cost. If your overall argument is that it is cheaper to kill people than to keep them in prison, you could argue that 1)we can streamline the process so we don’t have many appeals, 2) It costs too much to pay for nonproductive citizens in jail for life and 3) The cost of a death penalty trial does not need to be more than a regular trial since it is basically the same process.