It's unclear what you wish to define as overcooked in the case of stews because generally, overcooked suggests that there are unfavorable consequences, such as grilling chicken breasts for too long and it becoming dried out. When it comes to stew and meat in particular, cooking it or simmering it for longer periods of time won't necessarily overcook it, but rather determine how tender the meat will be. And, Since how soft/tender the meat desired is usually a personal preference, determining when it's overcooked for you is subjective.
In your case you mentioned the meat appeared "dyed out" which I'm not sure what you meant but i presume it just had an odd color or tint to it. Therefore, generally the only thing that you can really overcook would be any vegetables you've added (where they break apart in the stew).
To clarify, did you mean you simmered beef chuck roast (the type of meat cut used) or did you mean chuck *in* stew? In any case, depending on how you cut the meat up (size of cubes or pieces) will determine the needed cooking time until "well done." After well done, "overcooked" would apply to the various stages of how tender the meat got.
Based on the way you posed your question though, it sounds like you were hoping to cook it as quickly as possible since you want to know when to simmer longer or when it's overcooked (and implies when it's done). So if you don't need tender beef in your strew nor do you need to extract the beef flavor into the water and to more simply answer your question, you can just boil the meat instead of simmer it, and depending on the size of the cuts of meat, usually 5-10 minutes* at boiling will cook meat all the way through if less than an inch thick.
*the range of time depends on the temperature of the meat prior to entering the boiling water