Remember not to waste as it can and will either save you or get you kills.
is for the finishing blow. If you are 100% sure you can get the kill don't be afraid to ignite early as it will give you the extra AP and AD. However this can be a setback if the use an escape such as flash and you wont have your ignite up later.
& are also viable but used situationally. If you take these replace ignite, FLASH IS NOT CHANGEABLE (except for with ghost sometimes)!
Even can work, however I don't recommend or . If you really want, can work also in order to catch up to others for Shunpo's in team fights but I prefer flash.
Health/level SEALS: Help deny some auto attack harass in lane. If you don't have these you can feel free to take any of the other recommended seals but these are the best in most matchups.
MR GLYPHS: The MR helps you win trades in lane and deny a lot of incoming damage along with Katarina's base 1.25 MR/Lvl. If you really want you can take ap/lvl glyphs to get more damage but play careful as harass will do a lot.
M PEN MARKS: Standard, every AP mid lane champion should take these or hybrid pen but since you are melee take these.
QUINTESSENCES: Movement speed quints are really my favorite unless you think that you can win your lane easily and know you can get kills. The extra move speed helps you make plays in other lanes and help out your team while getting you an advantage yourself. The extra ability power if you choose to take the flat AP helps you do a decent amount of damage in lane however, the change is still quite minimal and against better players it is very hard to get kills in lane from just harassing. To be honest it is your preference that matters here because they are both a good option.
*Now i prefer running more ap and possibly 1-2 move speed quints depending on matchup.
21 in offensive tree for damage (standard). Feel free to go into defensive and utility tree or just utility if you want but for now I feel like the defensive tree is stronger. You can go 21/5/4 21/6/3 or 21/0/9 if you prefer.