Based on your character description, I will make some detailed assumptions:
1)Basically the living cells have become indestructible particles like atoms (under normal circumstances). I assume this power extends to all the tissues as a whole as well. Otherwise you can cut the tissue by simply separating the adjacent cells with sufficient force.
Effect: Character will start losing sense of pain slowly and steadily. The biological process and nerve reaction will still be there, but the brain will start ignoring the signals. The pain becomes a false trigger as there was no corresponding damage. On a long term, this might lead to desensitization and lack of empathy, slowly devolving into lack of certain emotions. This includes fear of death.
2) No cell division = no new cells at all
Effect: No reproductive capabilities may lead to degeneration of certain instincts, like losing interest in reproduction and related physiological and psychological phenomenon.
3) Despite being indestructible, the cells still need energy to function without which, the cells will enter a super dormant state, and wake up again after energy is available.
Effect: Most biological functions will continue. Character will experience hunger and thirst as well as the need to sleep due to getting tired. Combined with the above effects of losing some emotions and instincts, character may develop dependency on certain things. For a positive tone, this would be highly exhausting activities in terms of both physical and mental, otherwise character wouldn't need or be able to eat and sleep properly. For a negative tone, heavy addictions to junk food and/or drugs and other substances.
4) Cells do not die or divide, but the organs still produce the various chemicals, and body still accumulates certain substances over time
Effect: The character can easily "suffer" high blood pressure if consuming too much salt and fats primarily because the fat accumulates but the skin doesn't stretch and grow, kidney stone or other such problems caused by accumulation of external substances within the body. Of course these problems can easily be cured by strong medications easily since there should be no side-effects.
5) Long Life span
Effect: After a number of years, character will start suffering memory losses, because the brain has a maximum capacity for information.