Katarina is a melee champion that does most of her damage through spells. She is also very vulnerable to cc. As such she has better chances on top against bruisers and squishy ranged champs than on mid where mages are starting to become more utility-oriented, having at least one or two ccs. Katarina mid is still viable but harder to play.

Early, one should farm as much as possible. Constant harass is recommended but only when the enemy can be traded with. Once you hit level 4 you can start doing the q-e-w combo. Remember not to shumpo unless q hits and that mobs are better than harass. Just before level 6 try get your enemy low hp so you can surprise him with an ult when 6 pops. After 6 remember that your ult has a short cooldown so don't be cheap. If you can't get kills on your lane and the enemy just tower-hugs and you see an opportunity, gank mid, try help your jungler et cetera. By the time you're done your lane will be ready to clear.

A mistake I've seen so many people do is staying on lane with less than 30% hp. If the situation allows you to, recall. I've seen countless normals and low elo games where staying on lane with 300 hp lost an objective or prevented a team from getting one. Also remember that a kill is worth 17 mobs so forcing someone off the lane and farming 1-2 waves is almost as good as a kill.
The build here is in accord with my play style. As such, spell vamp is core, for it allows for the sustain kat desperately needs in order to maintain lane presence. Will of the ancients is nowadays trash on her considering it gives less ap, no ad, has no active and instead gives mana regen and a little cdr. The jungle item has recently been nerfed and as such it isn't worth it anymore. I negelected some items that are generally appreciated, like the abyssal scepter and zhonya and instead I've brought back the gunblade.
Regarding the Mercurial Scimitar, although it is weird, katarina has to benefit from all of its stats. She has AD ratios and that active can save you by releasing you from a deadly cc and making for a neat escape considering the additional movement speed bonus or it can turn a teamfight around. If the enemy has few ccs and spends one on you before you ult in a teamfight, you can get out of that and spin to win.