1 Perk in each skill means by lvl 19 you will be the absolute worst jack of all trades ever. hahah
I might end up doing that.

A mage that mostly specializes and uses destruction magic.
I'm not convinced that would be the
worst build. Surely maxing out Destruction is at least somewhat better than having part perks in Speech, Pickpocket, and Lockpicking.
Well, for the absolute worst character possible:
Lockpicking should be your first skill you dump all your perks in. This skill is absolutely worthless, since you can easily open any lock in the game without investing a single perk point.
(The only lock in the game that you will have a hard time opening without all the perks is the master practice lock in the thieves guild training room, since that lock resets its target zone with every pick you break.)
Pickpocket and Persuasion are also pretty damn worthless, though with 100 Pickpocket you could in theory steal someones weapons and armor and poison their pants before fighting them.
Your attack skill should definately be destruction, while it starts out the strongest, it ends up by far the weakest.
As far as clothing is concerned, robes are probaply the thing to go with. Robes in Skyrim have no advantage whatsoever, the only good thing about robes is that its relatively easy to get robes that carry a powerful enchantment from a vendor, but if you can make enchantments yourself you can easily get the same bonuses on any piece of armor. Of course your character won't be able to make enchantments because then he wouldn't be awful.
So, if we assume a character has 5 skills, then I'd say the most worthless combination of 5 skills would be Lockpicking, Pickpocket, Persuasion, Destruction, Restoration.
Restoration isn't really a crappy skill, but after the first 4 there is little to pick from that's really awful, and Restoration is absolutely worthless on a character with Destruction, since you'll be out of mana all the time.
Thanks for the suggestions.
The only thing I'm going to intentionally gimp is the perk selection process. The point is to simulate a player who plays well, but ends up picking a terrible build for whatever reason. That way, all other things being equal (making best armor/weapon/spell choices, using companions, being cautious, etc.) it will hopefully throw some light on the perks themselves. I'm going to do my best to play well and restrict my character's stupidity to choosing perks. I'll still be enchanting, smithing, etc., but I won't be grinding anything, I'll just be using them a reasonable amount. At present, I've only been able to invest a single perk in any skill because none of these skills are high enough to let me invest more. Based on playing smart and following the main quest, there isn't a lot of use for these skills yet. It's mostly combat, though my Sneak and Archery are pretty high since I'm playing it on Master and don't want to get annihilated every few minutes. I'm still not investing any perks in these skills though.
Out of curiosity, what do you think is the worst attribute to invest points in? So far I've put all my points into Stamina since Magicka and Health have obvious benefits and Stamina is a little more nebulous.