Thought I should throw in my two cents, since I used to play Leblanc quite a bit, and still do on occasion.
First off, you should have MR per level Glyphs. The blue ones, if I'm correct, for pretty much mid altogether. Also, health per level seals, the yellow ones, are good for mitigating burst damage.
During the early levels, she is going to Q harass and try to proc the debuff whenever possible. One choice is to try and exchange, pretty much don't be a *****. You'll be surprised how much damage you can actually pull off exchanging pre lvl 6, as leblanc's total burst revolves around her ult. Your other choice is to push the **** out of lane. Leblanc cannot farm effectively AND harass you at the same time. She either wastes mana securing cs, or throwing her spells at you. She can't do both. In my opinion, just play someone that can push the lane hard, push it out to her tower, then go farm wraiths or gank a lane. Leblanc will either have to sacrifice the wave you pushed to tower to follow you, or stay and farm at tower. Kat is decent in her Q-W combo in pushing the wave.
Now, that early burst damage by Leblanc is quite the *****. The best way to mitigate her early burst damage like that is actually health, and not MR. MR is good, yes, but buying an early giant's belt is better against Leblanc, since, as Kat, you are going to build a rylai's later, anyways. Also, going merc treads instead of your usual sorc boots will not only reduce Leblanc's overall damage, but shorten both the silence and chain snare.
Some other things to keep in mind, Leblanc is really strong during laning phase, as she can not only out-damage just about anyone she faces in lane, but also walk up and down lanes and put out gap-closing, large bursted ganks. However, Leblanc falls really hard once teams start grouping, as she wants to surprise targets that are singled out. Kind of hard to do that when the enemy team is grouped.
Another note, Kat isn't all that dominant in lane, which only widens the gap between these two. Katarina shines at ganking other lanes and teamfights, where her team can protect her from cc. Just remember to build a little tankier than usual, and you should be fine. Abyssal's, Rylai's, merc treads should be decent enough. A well timed Zhonya's should do the trick after that.
BTW if anybody has 20 stacks of ANYTHING, that is entirely your fault at making bad choices. The match-up doesn't matter. That is just being downright outplayed if anyone gets full stacks.