Consider how they age. Do they automatically stop aging at a certain point? Do they control their physical age? Keep in mind that physical and mental ages can differ, so don’t be afraid to experiment with this a little.
Don’t make them all knowing (unless they’re God or something similar). Just because you’ve lived a long time doesn’t mean you know everything. While having long lives does give immortal characters more time to accumulate knowledge, the world is always changing. New things are being discovered and invented all the time, so it’s impossible for a single person to know about absolutely everything. Consider having them be especially knowledgeable about a specific topic or two to showcase how they like to spend their ample lifespan, allow them to have cool specialized skills, and still allow them to have gaps in their knowledge.
Decide what can physically hurt them. Immortal characters may not be able to be killed, but they can often be harmed in one way or another. Can a mere mortal hurt them? Having them get hurt by something comically pathetic can make for some good comedy, especially if the immortal character seems formidable at first. If mortals can only hurt them with specific tools or with the aid of other immortal characters, then your immortal characters will be good sources of drama since they’ll be tough to deal with. Consider establishing that they can be incapacitated, perhaps in ways worse than death. Being chopped into dozens of pieces sucks, but it’s even more agonizing if you can’t die. Can you imagine being rendered helpless and incapable of communicating with others forever? Powerless rather than death will likely scare them more than anything, so use that to your advantage to add tension to immortal characters’ arcs.
Let things emotionally hurt them. Imagine watching your country fall into ruin or watching all your loved ones die while you remain physically unchanged. That sort of emotional trauma provides plenty of fuel for conflicts that immortal characters will inevitably have to face. Do they become resigned to watching things fall apart around them or will they try to do something?
Use them to explore big themes like what gives life meaning. Living for centuries can give you an interesting perspective on the world. Diving into an immortal character’s perspective and being exposed to their unique worldview can provide a lot of food for thought for the reader.