(covers information from several alternate timelines)
Rice attended Starfleet Academy with fellow cadet, William T. Riker. Riker remembered him as a very confident man, almost to the point of arrogance; he managed to get away with this trait because he had a tendency to be mostly right.
When Rice had taken his final test in Advanced Navigation at the Academy, there were three options available; Rice had ignored those three and had come up with his own solution. As a result, he had been awarded the highest score for the test, which was still in use at the Academy, but it now had four available solutions. Based on this profile, Counselor Deanna Troi reasoned that such a man would have a tendency to fight a battle, as opposed to retreat from one. (TNG: "The Arsenal of Freedom")
In 2364, Riker, now a Commander on the USS Hood, turned down a captaincy of the USS Drake in favor of continuing as a First officer with the USS Enterprise-D. Captain Rice served as commanding officer of the Drake instead. (TNG: "The Arsenal of Freedom")
Rice was on Deneb IV with Commander Riker prior to Riker's assignment to the Enterprise-D. In the anti-time past created by Q, Rice was seen during Jean-Luc Picard's communique with Riker, informing his future first officer the delay in the Enterprise's arrival to Deneb. (TNG: "All Good Things...")
Later in 2364, the Drake was assigned to Minos to discover why the planet had suddenly become unpopulated. The Drake was apparently destroyed by one of the weapons constructed by the people of that planet, a weapon which almost destroyed the Enterprise-D a short time later.
The same Echo Papa system that destroyed the Drake and killed Rice, would later create a holographic version of Rice as a ruse to attempt to extract information from Commander Riker. (TNG: "The Arsenal of Freedom")
Paul Rice was played by actor Marco Rodriguez.
Paul Rice also has a card in the Alternate Universe series of the Star Trek Customizable Card Game, which also contains an "easter egg" showing the words "daed si luap" ("Paul is dead" backwards) - a reference to the urban legend that Paul McCartney died in a car accident in 1966, and while The Beatles covered up his death and found a lookalike-soundalike replacement, numerous songs played backwards (as well as regular lyrics and images of the group) contained various clues revealing the truth.