Immortality. A blessing or a curse. The freedom of endless time and youth or being forced to witness everyone around you die over and over. The viewpoint of your character and how they act on that perception will be key to their story.
In many stories, immortality has limitations. Whether it be a certain kind of wood kills vampires, a silver bullet to the heart, or getting set on fire, most immortals do have a way they can be killed. You can extend this to things that will slow them down or hurt them. In Blood Fiercely Defended vampires can be burnt and broken like any other human but their healing is fast and gets faster as they age. Maybe your immortals are bullet proof but they can burn.
How someone would deal with immortality is a tricky question that will differ with each character. Some will see it as a gift, others a burden. Some will abuse their time. Others will feel they are obligated to do something with their time. Answering how your character will react will both be guided by their past and their community and could help you form their past and community. If you don’t have a character’s past tacked down, don’t fret. How they react through the story could help you discover what kind of past would best suit them/explain their behavior.
The rarity and origins of immortality vary. You must decide if this is something bestowed, passed down, or inherited. Perhaps it is a very recessive gene like in The Old Guard (great movie, if you haven’t seen it, watch it). Most vampire lore follows the characteristics of a virus, vampirism is bestowed from one vampire to the next whether the new host is willing or not. You could also take the angle that only so many immortals can exist at once. Perhaps there is a relic they must keep on them at all times. When this relic is passed from one person to the next, the first person loses their immortality.
Anyone that has lived a long life has made plane that how you keep going is purpose. You must give your immortals purpose. This could be simple; they wish to learn forever. This could be complex; they seek to save mankind from themselves. Without purpose, and with endless time, people drift. This drifting can result in hopelessness. If your immortal has experienced this you can start getting into heavy topics like depression and suicide. Handle these topics with care but don’t ignore them.
If your committed to trying to write immortality at accurately as possible I suggest reading articles from phycologists on the topic. Read first hand accounts of people that lived to be the oldest person in their town, village, or country. Witnessing immense change is a common theme. Feel free to explore this.
Ultimately how you write immortality, the rules you give it, are up to you. By setting down rules that your immortals must follow your writing becomes consistent. This can also lead to great plot twists. Explore your options and chase your curiosities. Don’t be afraid of creating something new and amazing.