There is an intelligent creature whose natural life span is around 50 years, during which they need to feed to not die of starvation, and after which they naturally die. But killing other beings of their own species increases their lifespan. By how much? It does not matter, but it needs to exceed their natural death of old age.
How can such a mechanism work in real life without magic?
Biological death clock
There are biological clocks you can implement in order to enable this to happen. Let's say, for instance, that these creatures have a unique brain which can only function in the presence of a rare metal to serve as a cofactor, say, zinc. However, zinc is not normally found in these creature's diet, so the way these creatures get zinc is very odd. The way I would do is make these creatures amphibious - they are born in the water and consume zinc-containing life forms there to finish developing to a land creature.
Now, these creatures will slowly use up zinc over the course of a lifetime and have no way of replenishing it under normal conditions. In order to keep their zinc from suffering from Le Chatelier's principle (that things tend towards an equilibrium), they have a special organ whose's sole job it is to keep surplus zinc. The organ fills up on zinc during the aquatic stage of life and can only hold so much.
But, should a creature find and devour another creature's zinc organ, then their own body can process that zinc and send it to it's own zinc organ. It'd get the remaining years of zinc, though obviously not all the zinc perfectly. At that point, it'd have added a few more years to that death clock, but it won't stop them from biological decay, so even with this, they'd still die from brain deterioration.
It should also be noted that once these creatures get to the level of modern medicine they can just make supplements and skip this whole issue. It should also be noted that I'm not really sure why the creatures would evolve to have all these very specific conditions.
Here are a couple of scenarios where I could see this working.
Your species is a parasite, and a nasty one at that. Your individual is out there sucking the life from it's host, killing the poor thing. Other members of your species are either gentler on your host (so they aren't dying) or else the parasite has been draining it for less time. Either way, your young parasite has a healthy host, and your old parasite has a sick one. Kill off the kid, and inherit the healthy host.
Your species is able to transfer their brains from old bodies to young ones. In humans, various stories have proposed this, usually involving younger clones of the person who are then sacrificed so the elder can have a healthy new body. If you want, your species has evolved a way to do this without modern science (immunology and interfaceable neurons, possibly females parthenogenetically cloning themselves then consuming their offspring).The hows of this would be workable but tricky. The whys? healthy new body.
Combine these two, and you have a parasitic species who have a life-long relation with a host. But Vlad the parasite wants to live forever, and kills a youngling, transfering itself to a young new body.
If your individuals are clonal, you could also have an individual killing a youngling for stem cells. Stem cells enable all sorts of healing and regeneration, but as we age, our bodies stop producing stem cells for various reasons. Your vampiric elder clone kills the younger, harvests it's bone marrow, and injects it into itself. Perhaps it wants to heal wounds that just won't heal. Maybe it has aplastic anemia and advanced medical knowledge. Either way, if you don't have a volunteer or advanced enough equipment, it's an evil way to stay alive.
