Okay, I've given this topic some thought of the last couple of hours. For me it's hard to narrow down a top 10 character list for a specific medium, let alone a comprehensive one. Still, I tried my best to pluck some good candidates from out of my head. I'm sure this is suffering from some recency bias and I'm also sure that I've probably overlooked some more deserving options, but this is what I've come up with
1) President Josiah 'Jed' Bartlet (West Wing)
The West Wing spanned 7 seasons and along with the quality, mostly consistent writing, I feel confident in placing Jed Barlet among my list of favorite characters, TV or otherwise. His character is a bit idealized, but the show does a good job in showing his fallibility while still emphasizing his remarkable qualities - his integrity, intellect, wit - to create an at least half-believable, wholly captivating character.
http://www.youtube.com/watch?v=dVgK5HKj3P4
2) Roy Batty (Blade Runner)
I've... seen things you people wouldn't believe... [laughs] Attack ships on fire off the shoulder of Orion I watched c-beams glitter in the dark near the Tannhäuser Gate. All those... moments... will be lost in time, like [coughs] tears... in... rain. Time... to die...
http://www.youtube.com/watch?v=a_saUN4j7Gw
3) Bob Harris and Charlotte (Lost in Translation)
This one might be cheating a little bit, but their characterization is largely developed through their interaction with one another and so I wanted to include them both together, rather than having to leave one out. They're both arguably assholes, but at their core, their characterization touches on the kind of loneliness that most people can relate to. I think it might be best to put that they come across as very humane.
http://www.youtube.com/watch?v=eNK7N3AwVFQ
4) The Boss (Metal Gear Solid 3: Snake Eater)
Man, those feels during that last fight and the following ending cut scene. She was a true... patriot.
5) Clementine (Telltale's The Walking Dead)
TWD was was one of the few times in a video game where I had to consciously deliberate on the choices I had to make, and probably one of the lesser than few times where the consequences left me feeling like shit. The game did a good job in instilling in you a sense of obligation to Clementine and the desire to protect her. The ambiguous choices you are forced to make and her horrifying reaction to some of them, well a video game character has never made me feel that way before.
6) Dae-Su Oh (Oldboy)
Revenge, revenge, revenge, revenge. Oldboy is probably my 2nd favorite revenge movie and what puts Dae-Su Oh on this list is just how badass his character is. Between the incredible hammer fight scene and the end he just ends up coming across as so striking.
7) Handsome Jack (Borderlands 2)
It was tough to pick between him or Glad0s, but in the end I feel Handsome Jack wins out. You hate him, yet you laugh so much you love him.
Niles Crane (Frasier)
I think that out of all the TV shows I've seen, I've never laughed as hard at a show than in the time I've spent watching Frasier. Partly of which can be attributed to Niles' character.
9) Rodion Romanovich Raskolnikov (Crime and Punishment)
An appalling execution of idealism, yet somehow you sympathize and maybe even root for him as he descends into madness and comes out of the other side redeemed.
10) Samantha (Gone Home)
A character you never see, yet however is focal point of the entire 'game'. As you gradually make your way throughout your home, you uncover more and more about your sisters life during your absence and I couldn't help but feel a connection to her and neither could I help from feeling a growing concern about her fate.