I look at overall premise of the character (background, concept, motivations, etc.) as much as I do other aspects. What makes him/her unique? What makes him/her interesting as opposed to all the other potential characters out there? What drives this character to do what he/she does? While these questions don't necessarily give any technical advantages in terms of game stats, they do help provide insight into the character and as a result the role-playing experience is more three dimensional and a LOT more fun.
To illustrate, allow me to put forth my two favorite HU characters of all time -- incidentally, which are ones I'm currently playing (well, sort of anyway).
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William Blake, a.k.a. PROWLER
SUPER ABILITIES: Animal Abilities (Big Cat), Heightened Sense Of Smell, Heightened Sense Of Hearing, and Wingless Flight
PHYSICAL CHARACTERISTIC: Retractable Claws
On the surface he follows a pretty cliched theme -- the newspaper reporter/photographer by day and the superhero at night (though he's just joined the U.S. Marshals and is about to be a crime-fighter full time). However, what makes him really interesting to me is that he tends to do a lot of investigating, and that's where Blake's powers come in the most useful (following scent trails, overhearing conversations, hovering outside skyscraper windows, etc.). As a result of his continuous investigations, he's one of the most well-informed characters in the whole campaign. Also, while he isn't the most powerful character in the game he still manages to come out on top or at least hold his own no matter the opponent. I guess it's his versatility in both combat and non-combat situations that makes him fun to play.
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Kalok "Kal" Bonner, a.k.a. STAMPEDE
Minotaur Mega-Hero Super Soldier
SUPER ABILITIES: Extraordinary Speed, Enhanced Leaping Ability, Bookworm, and Heightened Sense Of Awareness
SUPER SOLDIER ENHANCEMENTS: Brain Boost, Increased Physical Agility/Dexterity, Mind/Body More Attuned To Each Other, and Uncanny Targeting/Throwing
MEGA-HERO POWERS: Tremendous Strength and Tremendous SDC
MEGA-HERO WEAKNESS: Extremely Nearsighted
Yep, Kal is by far the furthest I've ever travelled down Munchkin Lane, but even so he hasn't been all that unbalancing during game play. Kal started out as an idea for a Rifts campaign (an idea by Tolkeen to help better their odds against he Coalition), but was brought into HU under the premise that minotaurs and certain other 'mythical' creatures are in fact alive in the same sense as Bigfoot and the Yeti (basically a case study in cryptozoology).
His premise is that he was mentored by a homeless ex-soldier, kidnapped by an unscrupulous company and experimented upon, and in the process became an 'information junkie.' What makes him interesting for me to play is that while he didn't ask for these powers, he relishes his newfound intelligence and spends most of his time reading and has been known to spout quotes from obscure poets in the midst of battle. He's not very skilled (so he veers away from the typical 'smart brick' cliche somewhat), but don't sit down to a card game against him. In short, I play him for the new brains at least as much as I do for the new brawn.