First, start with a elixir of Intelligence and at least three health pots. Without runes or Masteries it gives you the best AP-ratio of all starting Items and 10% cd reduction to boot. Combined with the 600 hp from the pots it means you will outperform anyone else in the first few levels.
If you get a kill, and you will get one, trust me... The elixir pays for itself and then some, as well as you have 125 gold+ saved from the very beginning. Buy yourself boots right afterwards and go for a fast Rylais to boost your Ulti ( It slows the enemies so the full Death Lotus hits them). Then upgrade your boots and get a Radabons Deathcap. The Banshees veil is there for you not to be interrupted during your ulti. Afterwards, if you are pwning hard, get a Mejais to snowball even harder and to crush their Morale. If you are in a tight spot before a Teamfight, consider the extra 60 ap from another elixir.

If the Game is still not over, i suggest getting a defensive AP items.
Congratulations, you are now ruling a low levels with Katarina.

Have fun and reccomend my Guide.