As others have pointed out, it won't work like in Plague Inc: If a relatively harmless virus spreads through the population, and somehow evolves in one place to become lethal, then the lethal version has to spread again, the harmless viruses will not suddenly all change to become lethal.
However, it is possible to have a virus that first infects everyone and then, later, kills many people. It probably won't infect 100% of the population nor kill 100% of those it infects. But it could affect a large enough part of the population to end society.
The evolution of such a virus without a human designer is very unlikely, but not impossible. Here are some options for how it could work:
Long Term Effects
The initial infection with the virus is harmless and only causes whatever symptoms are needed to spread the virus. But in the long term, either the infection damages the body so thoroughly that it becomes difficult to survive, or it causes some secondary effect (through toxins, destroying tissue, overstimulating the immune system...) that kills the host.
It could take many years between the initial infection and the secondary damage, so that people won't discover the connection and won't take drastic measures to avoid the virus's spread before it's too late.
A real-world example is HIV. The reason it didn't infect everyone is because infection can only be spread by intimate contact or medical procedures. So humanity as a whole had time to find medication that can render the infection relatively harmless. If the infection was spread easily by casual contact, things would be quite different.
In this scenario, it would almost have to be a virus, because bacteria or fungi are generally easy to kill and therefore are unlikely to remain active in the body for very long. Bacterial infections generally aren't as contagious as viral infections, either. However, they could in principle kick off some other process in the body, that ends up killing the host when the bacteria are long gone. For example, they could produce harmful prions somewhere in the body, that cause nerve degeneration years later. In this situation, the connection between the infection and the deadly effect would be especially hard to establish.
A Secondary Trigger
Again, the virus infection itself is harmless, but the virus stays active in the host's body for some time (months or years) and ends up killing the host once some secondary stimulus is introduced. This would have to be something that happens to everyone, but not often. It also needs to be something that can't easily be avoided: If people start dropping dead after swimming in a pool or getting anaesthesia, those things just won't be done any more.
Maybe the virus has a deadly interaction with the common cold. Or it makes the body unable to tolerate heat, cold, summer sunlight. Or it causes a deadly allergy to some common pollen: All these things won't show up for a few months if you time it right. Various science-fiction triggers can also fill this role.
For this scenario, your infectious agent could be anything. Virus, bacteria, even some environmental chemical (though you'd have to explain how it spreads to everyone). And since the trigger can be something that everyone is exposed to at roughly the same time, you could have the sudden effect that you want. However, the death rate will probably be lower than in the first scenario, because there will always be groups of people who manage to avoid either the infection or the secondary trigger.
There's one other thing that makes reality different from Plague Inc: Just because people notice that a virus exists and may be connected to some deaths, that doesn't necessarily mean they'll find a cure in any specific timeframe. So your virus doesn't have to be undetectable to have the intended effect, it just needs to be able to spread in a hard-to-avoid manner.