Depends on your tastes, but a few Netflix shows that I enjoy a lot and have good characters:
Fringe (if you go with Fringe, bear with it. Takes about half a season to find its legs, but you won't be sorry. Plus, Walter Bishop is the best).
Freaks and Geeks - Amazing high school dramady, and basically everyone in the cast went on to do good work so you get the gratification of seeing the beginnings of some pretty amazing careers. Great show.
Twin Peaks - Totally weird sort of crime soap opera by David Lynch. Surreal and fun and cool.
Parks and Recreation and Raising Hope are really good sitcoms that really respect and care for their characters. If you watch Parks and Rec, either go straight to season 2 or be ready to kind of bear with it through the first season. It became pretty much my favorite sitcom of all time, but it took some time to find itself.
The West Wing is definitely worth watching as well. If after 4 or 5 seasons or so you start feeling an itch to bail out on it you may want to. I liked it all the way through, but at a certain point after Sorkin left it wasn't the same. Great show though.
Doctor Who
Scrubs
How I met your mother
Dexter
Supernatural
Firefly
Battlestar Galactica
Sword Art Online
Breaking Bad
Orange is the new Black
House of Cards
Here are some shows that I didn't like because, well, I'm not a girl. But I did watch them with my ex gf, and I could see the appeal.
Pretty Little Liars
Scandal
Revenge
The Secret Circle
Nip / Tuck
Since you did say specifically "not a comedy" Breaking Bad & The Walking Dead would be my top picks. For comedy with good story and likable characters: How I Met Your Mother & Parks and Recreation.
Switched at Birth, Walking Dead, Always Sunny in Phili, Trailer Park Boys, Family Guy, Futurama, Breaking Bad
Harper's Island was good for it's "whodunit" type of show, and when you discover who the murderer really is....a bit of a shocker.
I'm looking for something new to watch, not a comedy, something I can really get into, with a good story, and characters you really feel for
