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I understand that in D&D 5e, there is a certain age range that characters can be. And we know that old age is probably something that will happen to most characters after an adventure. But occasionally, it may happen during an adventure.
How do I determine when a character dies of natural old age?
I have not reviewed every 5e book, but I have several of them, and to the best of my knowledge, this is never addressed in any of them. More than that, I don't expect it to be addressed in any of them in any detail because this is simply not the type of thing that tends to get described in a lot of detail in 5e.
The fact that characters can die of old age is mentioned many times, and the average life-spans of many fantasy races are mentioned, but the details of aging and thus an answer to this question is not something likely to be covered in much detail in books about high adventure.
(Magical aging is a separate thing and is discussed multiple times. But in general the specifics when magical aging are involved are usually discussed alongside the effects).
I have a hard time coming up with a scenario where you would want to roll for this. Instead, if it becomes relevant, I recommend letting the story dictate it. If it is dramatically appropriate in story for a character to die of old age, and they are not so young that it would be ludicrous, then they do (Think Yoda in ROTJ. The timing was very convenient story wise.) If the story would be served by them living, and they are not ludicrously old for their species, then they do not die of old age (and if they are ludicrously old, but their death would be inconvenient, there are various magics that could be invoked to buy them a little more time while maintaining verisimilitude).
If you really want to use dice for this, then at least for humans you can look to real world actuarial tables like this one.
There is a death probability chart for each age. You can roll a percentage (use a d100 or 2d10 with one of them representing the tens place and the other the ones place). If they beat the odds, they won't die of old age for another year. If they don't beat the odds, they will die at some point in that year.
This of course is not perfect because most actuarial tables look at death from any cause rather than just old age (It can in fact be difficult in real life to say whether someone died of "old age" or not. If someone died from an identifiable disease that a younger person probably would have survived, did they die from old age?) Still, it makes a decent starting point if you want to roll.
You may not want to use a constitution modifier at all. For one thing, 5e at least to the best of my knowledge does not have rules for reducing constitution with age, which is an expected result of aging in the real world (though how much gets complicated and depends on how well someone has been taking care of themselves, genetics, environmental factors, etc). I would simply disregarding constitution. But if you want to factor it in while using the actuarial tables, consider adding the entire constitution number to the percentage roll. This could result in some people living exceptionally long lives, but that might be expected at least of people like adventurers that are meant to be exceptional.
If you're willing to cast the net a bit further back in time, 3/3.5e D&D tends to go into much more mechanical detail on subjects that 5e abstracts or ignores, and ageing is no exception; with respect to maximum age, the rules for ageing state:
When a character reaches venerable age, secretly roll his or her maximum age, which is the number from the Venerable column on Table: Aging Effects plus the result of the dice roll indicated on the Maximum Age column on that table, and records the result, which the player does not know. A character who reaches his or her maximum age dies of old age at some time during the following year.
For instance, humans are given as "venerable" at 70 and having a maximum age of +2d20 years. So, barring an early demise of unnatural causes (to include disease and infection), or a postponed demise due to magic or some other sort of intervention, your common-or-garden human will die somewhere after turning 72 and before turning 111 - on average, at 91 years old. The DM knows what year it will be and the exact timing is left up to them.
The maximum ages described in 3/3.5e should roughly align with the expected lifespans described for races in 5e, so you could probably use these rules directly without much issue - at least where the two editions have races in common, and coming up with sensible formulas for those which aren't covered in this much detail isn't difficult.
What kills people isn't old age; it's organs wearing out or malfunctioning, or disease or toxins, or some other specific problem. Age just gives time for these insults to accumulate and for the body to lose its ability to recover from them.
Modern medicine can greatly extend life, even when not directly aimed at doing so. In the fantasy milieu, magical/clerical healing can probably have a similar effect. Exercise also helps, and most PCs are getting plenty of exercise...
Generally, we assume that most characters -- rushing directly into danger as they tend to do -- will be killed by something more obvious before suffering a "natural" death.
And most players will be Really Annoyed with the gamemaster if the character has a sudden heart attack or stroke and they can't immediately be healed at least enough to get back to town and find a more powerful healer, unless you prepare them properly by explicitly having the cleric discover and cure them of diseases earlier in their lives so that's part of their expectations. If you're going to introduce this variable, do it right, or you're going to break gameplay.
(I've always thought that if clerics get to the end of the day without using all their prayer points, they should do health checks on themselves and the rest of the party and heal anything they find. If you want realism, you've got parasites and sprained muscles and disease and lots of other things that could reduce the party's effectiveness. Don't try to tell me that party members getting bitten on a regular basis don't risk rabies as well as lycanthropy -- and rabies is nasty even with modern treatments; it can cause agonizing death in a matter of days and by the time recognizable symptoms occur it may be too late to treat with less than heroic measures. Presumably in a magic world it can be cured, but first it has to be recognized...)
We gloss over a LOT of real-world issues in the usual campaign because it's hard to make them fun. If you're willing to do the additional work to establish them as part of your world and find ways to make them enhance play rather than hinder it, go for it. It's not that different from introducing inflation in prices, or carrying-weight limits or other more-real-world issues; it can enrich the story if used well but it comes with a lot more workload for the GM. Reality is complicated.
Afterthought:
If you're going to do this, you should also be considering nonfatal aging. Does the hunk with extremely high natural strength start losing some of those points as they get elderly? Does Constitution decline? Dexterity (though that's offset by gaining skill)? Does charisma depend more on player's willingness to play the character as charming/benevolent? The game can become arbitrarily complex, but it has to be something the players and gamemaster can intuit and act upon in near-realtime to be playable.
