Well, a few things define a great character. The popularity is something else entirely. Some important aspects of a character, I would say, would include the following:
~ A very well developed history and personality, meaning: The character's personality coincide and aren't just random words placed together. They should have a lot of thought put into them, not just be thrown together so you can have another character.
~ Try to be original in breeds, names, designs, etc. When it comes down to it, a lot of people like unique characters that aren't just another drop of water in the ocean of other characters. A very good character usually has an uncommon name, or a few different breeds not many on the site have used, and of course, the design can't be referenced from anyone else's. It should be your own sort of idea. You could always, of course, ask other to do the design for you, but I find people tend to like the characters even more if you designed them yourself. As for the designs being complicated... well some people prefer simple and others very complicated. So a good median is prolly best. However, it's your character so therefore, your choice.
A few things on popularity?
Well, if you're naturally popular on AS, of course most of your characters are going to be popular as well. But just one way of getting your characters known is the
"Rate Your Character" topic on Forum Games. There you can rate other people's characters, post your own, and then someone else will rate yours.
Also, the elements that make a good character should be there. Otherwise, I doubt many people will want to see them. Oh and one more thing? Your characters do not need to be sparkle dogs or have unnatural personalities in order to be interesting. A natural looking character can be interesting all on their own. Granted, sparkle dogs or unnatural dogs are probably a bit more fascinating, being that they're not normal, but then again, not many people I know are too fond of them.... (I think they're cool, but not many others do from what I can see...)
That's all I have to say ^^ Hope it helped any :3