Jack wants one thing in his life. He wants the Doctor. And not in any sexual way, though that would be nice. He wants the Doctor to realize that he'll always be there, even long after the previous companions and future companions are gone.
He knows the Doctor would never consider taking him on as a permanent companion, due to the whole "immortal" thing. It's not just that Jack's existence is everything but right, its the fact that if this were to happen, the Doctor would watch him put his life out there to protect him time and time again. And Jack would never know how much that would hurt him.
Jack believes the Doctor would never accept him only because of his immortality, and because he believes he's ruined the Captain's life by being involved in the incident which caused the Immortality. But Jack, oh to Jack, the Doctor has done nothing but good. He's given him the ability to help people until he no longer has to, and in a sense made it easier to relate to the Doctor as he himself watches those he love die constantly from different causes.
It's not uncommon for Jack to wonder what life would be like if he could travel forever with that madman in his little blue box. He wondered how much better life would be if he got to see the Time Lord every day until the very end of time itself. But he knows that's just a silly dream.
The Doctor could never accept Jack, or so he believes. So every day, when Jack finds himself thinking of the Doctor, he tries to distract himself with other things. Though this never works, and it never will.
On those rare occasions he sees that police box parked somewhere in the distance or just blending in with the streets, it's always after he's almost suppressed the feelings he holds very deep inside himself. And just like that, all the walls, all the barriers, they come crashing down on him and he's left with that same vague emptiness that's only filled when he gets to see the Time Lord on some chaotic or peaceful journey.
As the days stretch on, Jack finds himself growing more and more impatient and feeling even more abandoned by the man. However, how does one feel abandoned by someone they only see once every few decades?
On the days things feel particularly hard to bare, Jack has to remind himself to push through it all. Just like the Doctor would tell him to. Because that's how his life has been changed by that daft alien in a flying blue box.
Even when he feels down, abandoned, defeated, hated, alone…he stays strong and keeps a smile on his face. Because he knows the Time Lord, no matter where or when he may be, is always watching. Even if only in his memory.