This lane is really snowbally. Basically, whoever gets an early lead should be able to use that to crush their opponent. I would honestly recommend putting several points in W in this lane (maybe 3) before you start maxing E because you are both melee. After level 6, you should probably try and save your E to avoid her ultimate's damage if possible.
Katarina will likely purchase one of two items: Deathfire Grasp or Abyssal Scepter. If she gets DFG, be very careful about approaching her at anything less than 3/4 health, since she can easily burst you if you don't avoid her ultimate. In this situation, I would recommend just pushing the wave, then backing off to roam or clear jungle camps between waves. If she buys Abyssal Scepter, you will have trouble bursting her so poke her down beforehand. Sheen is a good rush item against Abyssal, as it gives you physical damage.