(I am currently experimenting with builds and am having a hard time deciding on a good one, what follows is an explanation of the above build without rylai's crystal scepter, however i am coming to realize that rylai's is an almost must have on kat so I am continueing to experiment)

I know there are a lot of actives to keep track of but at kat's current state she really needs some of them.

Full combo with this build
Flash in (optional), DFG, Hextech Gunblade, W, shunpo (to be as close as possible), Ult, Q, AA/Q/Shunpo till dead

With the amount of AP from this build DFG will be doing 41% of their current health as magic damage seeing as kat has an unreliable ult this is a nice addition.
Math: 25% + 4% (per 100 ap) * 4 (464 ap /100 round down) = 25% + 16% = 41%

Hextech Gunblade provides a much needed slow and should be used before you use her ult to keep them in range long enough to get full damage.

Kat gets focused hard especially when her ult is cast so it is suggested to A. stay back until others initiate B. always use W before shunpo to increase survivability, C. When you start getting focused pop Zhonya's even if that means cutting your ult short. usually if you pop zhonya's during a team fight it disrupts the enemies that are focusing you for at least a couple seconds for them to aquire a new target. And during team fights if you pop it near the begining you will be getting at least an assist while it is up therefore reducing all your cooldowns by 15 seconds so you can come out of it shunpoing.

voidstaff vs abyssal scepter: The magic penetration from Voidstaff outperforms abyssal scepter when the enemy has above 50MR. if you have a lot of team members doing magic damage and need the MR yourself you may opt for abyssal scepter.
Math: .4 * X = 20
X = 50 so at 50 MR voidstaff is giving the same as abyssal scepter

Rabadons + Voidstaff = lots of AP & MPen -> More Damage -> Faster kills -> more cooldown reduction from passive -> win

Build order after Hextech should be adjusted for what you need, suggestions:
If getting focused hard -> zhonya's
if they build lots of MR -> void staff
DFG before Rabadons unless you feel that you are doing well enough without DFG and have built another item after hextech

I always have a hard time choosing two of these because there are 3 that fit great.
I usually end up taking exhaust as I see it as one of the most useful summoner spells.

Exhaust/Ignite - really good for early game kills and countering AD carries and Health Regen champs

Exhaust/Flash - your enemies will never get away from you, and you can also use for escapes

Ignite/Flash - good for tagging those enemies that are getting away with low health, provides both mobility and damage

Individually:
Exhaust - A great spell for Kat. the slow can keep your enemies in your ult longer and with the mastery addition of reducing their armor/MR you will be doing more damage to them. along with them doing less damage to you.

Ignite - very nice early game to help deal damage, use it first when you go in for a kill to amp your ad/ap by 5, early game that will help, and is less important late game


Flash - nice way to get in and out while A. shunpo is on cooldown and B. you want to save shunpo for something else (flash shunpo to catch up or shunpo in flash out harass/towerdive)

AP Nukers will eat you, you have relatively low health and very little magic resist, if an annie or veigar is within range and you are in their sites you will most likely be killed.

This is why I try to never go mid with this build. mid lane is filled with ap nukers. This build does better when you are able to keep to the bush's and avoid damage until you decide to jump on someones face and kill them.