1. The Doctor is 900+ years old. This is possibly the most important thing about the Doctor that makes it exceptionally difficult to play him. He's very, very old. He's seen civilizations start up and end just during his lifetime. The language of the people he met when he was 30 isn't spoken anymore. If every year showed he'd look like a cross between Gollum and Gir. He holds a very childlike innocence in regards to exploration, travel, and romance, but there's very little that he's not done. I have to remember to keep that aspect of him solid without making him too jaded.
2. The Doctor is the sum of his parts. I think it is impossible to accurately depict the Doctor in any incarnation unless you are familiar with the former incarnations. As the Fifth Doctor said in Five Doctors: "A man is the sum of his memories, you know, a Time Lord even more so." No matter how different the Tenth Doctor is from, say, the Third Doctor, that velvet-wearing, karate-chopping bloke is still inside of him. He even name-drops like the Third Doctor does. Not every Tenth Doctor can be played by a rabid watched-every-existing-episode-in-their-y
3. The Doctor will never settle down. There was nothing more terrible to the Doctor than being exiled and unable to travel in series 8. Why? Because he's a born explorer; a nomad. Even if happiness could be served to him on a silver platter, if it required staying put in one place too long, he wouldn't take it. That's what makes the Doctor so different from John Smith. John Smith wanted the slow path, the Doctor wants freedom. He's like a shark. If a shark stops swimming, it will die. No matter the situation, the Doctor can not stop trying to escape.
4. The Doctor can not become too serious. That's not to say that he can't be serious, that he can't get angry. Of course he can. It's the stab of irony ("You can't even sink the Titanic!"), the twisted sentiment ("Quite right, too.") , and the bloated ego ("I'd call you a genius, but I'm in the room.") that make a simple angry/heartbreaking/emotional scene into a scene that's about the Doctor. It's a very difficult side to him that can't be overdone (see: Tooth and Claw) or underdone (see: Last of the Time Lords) at the risk of making a serious situation seem silly or a dramatic scene seem sappy. So, when in doubt, I throw in some situational humor ("Can't believe it took me that long to reverse the polarity!") or a completely inappropriate reference ("No, wait, that's the Lion King.").
5. The Doctor is a hero. No matter the situation, no matter how dark I write him, I have to always remember that the Doctor is the childhood hero. He always fights for what is right. Not always what is good, or beautiful, but what is right. If he's not a hero, he's just some bloke played by David Tennant.