If there is one conclusion that is shared among the large number of research publications and game development books & articles investigating this subject, it's the following.
The process of creating gaming experiences is complex and associated with a host of variables that contribute to greater or lesser extents, depending on the specific situation, player and game. This is also one of the major reasons why experience is vital to a designer of player characters.
Stats are, of course, the rules based component, and something we see creatively used in games today. Integration is more rarely exploited, and the degree varies.
Usually the background for a character is not well developed in FPSes or CRPGs, with Neverwinter Nights a good example - we just know that he/she is a hero in training come to aid the city. Other games like Star Wars: Knights of the Old Republic utilize background as the premise for the entire game story.
Personality is an area which is more rarely utilized, possibly due to the existing "blank slate" paradigm. There are some exceptions - KOTOR has an alignment system, Call of Cthulhu: Dark Corners of the Earth has an insanity system, the Final Fantasy series generally has pretty well-rounded characters, and most D&D games integrate an alignment system.
These are all relatively straightforward compared to what is possible in tabletop RPGs, even when alignment affects NPC reaction. Furthermore often players are served the personalities of their characters in a passive manner, e.g. via the way their character behaves during cutscenes (e.g. Beyond Good and Evil).
Role playing communities tied to specific games, e.g. Neverwinter Nights or RP-guilds in MMORPGs, offer an interesting contrast - here players often have thorough descriptions of their characters and utilize these.
In the persistent games, the RP guilds do a lot to integrate them. This would seem one of the main reasons why people are still hanging in there. In other words, this is a strong driving factor in player retention.
In tabletop RPGs, the players need to create integration and personality from scratch. Some, such as GURPS, Mutant Chronicles, Traveller, or Vampire the Masquerade, provide a more or less developed system for assisting players in creating these features with a rule-based component.
For example, acquiring a good connection in the government costs two character points. Being a pacifist gives you two extra points (apparently a disadvantage!), and you may take up to 20 points in disadvantages.
Such systems do not replace the full image that players can and generally do build of their characters, but for computer games provide a direct path to integrate such systems because they are rules-based.
Personality-based character elements can be integrated in various ways into the overall game structure. In general, we can categorize a complex character system based on its depth - i.e. how detailed and complex the system is - and the level to which it is integrated into the game in question.
For example, a character personality system could have a rules-based effect, be used to control content access, or have no in-game effect at all, being primarily a tool for the player community. We can even integrate a personality system that operates outside the game mechanics but features its own rules for character development.
Depth of system: In general, we can separate between a shallow and possibly mechanistic approach, where personality/integration elements of the characters are chosen or generated from a simple system, or a deep system where the effects are embedded in several elements of the game, e.g. mechanics, story, and other content.
A shallow approach could be a list of psyche elements to choose from, with each element leading to a simple rules-based effect. Shallow approaches are not better or worse than deep approaches, but each offer specific benefits and have different resource requirements.