I don't know if the Trooper has to get certified copies or not, but the process in each county is not difficult in those that are computerized.
I have done such checks in Dallas, Tarrant, Denton, Collin, Travis, Bexar, Harris, etc.,. There are public access terminals and you can check your own. however, to get an official clearance document or proof of convictions, you have to see a clerk who does it. It is a same day process.
In counties that are not computerized (and I have been in a ton of them) the procedure is much more tedious, but it is a same day process.
I don't know what the work load is, but I imagine if there are many names to be checked when the Trooper goes in, he could leave his list with the clerk and return another day when they complete the list.
I also don't know what other duties the Trooper may have. I am sure that there are not 254 Troopers (that the number of counties in Texas) assigned to nothing but CHL duties. They probably assign the Troopers according to the DPS regions. Each DPS region encompasses many counties. Here is a region map;
http://www.txdps.state.tx.us/administra ... es/map.htm
Each region usually has 2 districts indicated by A or B etc, within each region. Troopers may be assigned to districts, which would be a lot of area for a Trooper to cover. Each district has a number of Sgts, depending on the size of the district and number of counties. They could be assigned to each SGT of a District who assigns them to a Trooper.
Of course, the backgrounds could also be done by Criminal Intelligence, (they are criminal investigators whose function is the gathering of criminal intelligence) who are assigned per region.
At any rate, DPS does not have people assigned strictly to the CHL backgrounds. Those checks would have to be done in conjunction with with the person's other duties.
IIRC, and Charles can correct me if I am wrong, DPS received no additional funding to manage the CHL program.
EDIT to add' Oh, And I don't know if DPS has a state wide policy on who actually does the in county checks. The "Director's Designee" could be the Capt of each region, who decides on a region to region basis who does the checks.
Unfortunately the CHL laws nor the Texas Administrative Code are more specific in who the "director's designee" is, or how DPS is to accomplish the required background.