One can follow p53 protein levels by immunoblotting in total cell extracts. The p53 protein is stabilized by modifications on certain amino acids. Therefore, depending upon the cell system (mouse/human; cell-type etc), one can follow the modifications too.
This is a tricky question that need more information as it will depend whether you want to study the expression of an endogenous TP53 or exogenous by transfection. What do you mean by stability ? You want to define the haslf life of the protein or just check the level after a specific stimulus. With more info, we can be more helpful.