I got the leather jacket and Sulik. In fact, I got the leather jacket from Sulik.
I think there also may have been a glitch because I got training from that dude to skin pelts, but it won't give me the perk and I can't skin the geckos.
A leather jacket is not the same as leather armor.
Leather armor, in your inventory, is brown and only shows a chestplate, easily definable by the pectoral chest mold. You can usually trade for one from the chic that has Sulik working for her, and some times you can even get it from the bartender in the other bar. Occasionally he'll have it, and he's cheaper to barter with.
As for the gecko skinning...
You went north to the caves to find the dude, right? Once escorting him to the exit of the caves he will meet you back at the same bar where you find Sulik and teach you.
I can't remember names at the moment. Sorry?
Also, don't forget to clean out Trapper Town too.
And once you have the Gecko Skinning perk, go to the green map exchange on the southern Klamath border and find the hunting grounds or whatever and you can kill and skin all those geckos. The pelts become a bit heavy in bulk but you have Sulik to mule them around for you until you're ready to sell them.