The personal armor of Joshua Graham consists of a white collared shirt with rolled-up sleeves embroidered with Dead Horses tribal markings, heavily patched blue jeans with a hole in the left knee, rattlesnake belt with dual pistol magazine pouches and a pre-War Salt Lake City police department SWAT ballistic vest.
Additionally, the armor has a sleeve garter on the strong/firing hand to ensure the sleeve doesn't distract the user. Completing the outfit are a snakeskin boots and wrapped fingerless gloves.
It is found in the footlocker that the player character appears in front of at the end of Honest Hearts, or it can be taken from Joshua Graham's corpse if he is killed.
While wearing the armor, the Courier does not receive the bandages that Graham wears. This is because Graham wears a non-player character only version of the armor with the bandages included. Due to this, only the player character version of the armor can be looted from Graham's corpse, but he will still wear the non-player character version. Both versions, however, have the bandages on the knee.
Joshua Graham's unique non-player character version of this armor has a DT of 40, which combined with the DT of 10 from his headwrap gives him a combined DT of 50.
In the intro slide show for Honest Hearts, Joshua Graham is shown wearing this armor even while he was a Legate.
According to Josh E. Sawyer, "Lanius' armor seemed inappropriate for Graham. Though it's unlikely that Joshua would have worn the same clothes then that he does when you meet him in Honest Hearts, there weren't a lot of other appropriate clothes for him and his outfit does make him stand out as particularly unusual -- which, even among the Legion, he was."
Wearing this clothing does not have any effect/special dialogue while talking to Caesar or Legate Lanius.
Behind the scenes
SLCPD stands for "Salt Lake City Police Department" and SWAT is an acronym for special weapons and tactics. Joshua himself never mentions how he got his armor.