BlueMooner: As others have said, it depends on how your define CRPG.
I grew up playing actual RPGs, games like Dungeons and Dragons. It was only after RPGs were popular and computer gaming rose up that we got CRPGs. I think this history is important because I see some gamers today defining CRPGs as having to do with levelling and stats. That to me is completely wrong.
D&D had levels and stats, sure, but D&D was first and foremost about story. It was about having an exciting campaign with interesting characters in exotic locations or situations and, most importantly, a story that you could shape. To me, what makes something a CRPG is a game with a great story and detailed plot, a plot that you can alter by your choices, and combat. VTMB and Deus Ex didn't have levels, and they're both great examples IMO. They also had choices you could make along the way leading to different endings. Stats and levels are immaterial.
Point N Clicks (PNCs) aren't RPGs because while there is a story, you can't change any of it. It's basically a visual novel with tons of clicking. Story, plot, choices with significant consequences.... these to me are what makes a CRPG. Fallout had a great story, BG had a great story Gothic had a great story. All had stories, all had choices. Combat is there to keep you from getting bored with all the talking and reading, and all the talking and reading is there to make the combat meaningful and significant. If you're killing generic enemies and just fighting for no plot reason (aka grinding) then that's not a CRPG element, that's an FPS element.
The whole aspect of the "role-playing" in the name was the ability to create a character of your imagination... kind or cruel, polite or trollish, pugnacious or pacifistic. Obviously programming can only offer so many options, but being able to play a character YOU want in a game, instead of a pre-defined one, is inherent to the genre. In all the games above, you have options on how to shape your character, both in combat and dialog.
The way I see it, CRPGs are not defined in the same manner as TRPGs (the T stands for "Tabletop" here). Story does not make a video game an RPG; many other genres, like visual novels, have stories. In fact, the game I am playing right now (Gurumin: A Monstrous Adventure) is an action game, but yet it has as much story as a typical JRPG.
Also, a game actually needs to be in first person and have you wield a ranged weapon for it to be classified as an FPS. Killing enemies and just fighting for no reason can happen in games that are clearly not FPSes, such as the original Dragon Quest (rpg-style stat based combat) and Zelda 2 (sidescroller once combat has started). So no, that doesn't define it as an FPS.
In any case, the way I see it when it comes to video games:
* Story, particularly when told through cutscenes, is a visual novel element.
* Being able to alter said plot is, again, a visual novel element (and is what separates visual novels from kenetic novels, which are pure cutscene with no branching).
* The one aspect that is actually unique (or at least almost unique) to RPGs and isn't found in other video game genres is turn-based stat-based combat. The only other genre to have anything else like it is the turn-based strategy genre, and there a unit's stats are typically fixed, unlike in an RPG.
Remember that games like Wizardry 1 and Rogue are generally considered RPGs, and the plot in those two games are really just excuse plots.