The main goal aims to build a viable Katarina. This means most possible damage output (raw AP plus the needed magic penetration) whilst longest survivability due to some vital defense and sustain through some spellvamp.
Further some cooldown reduction is useful to poke more often in early game and chain your combos late game to a devastating dance of death.

Quintessences: Cooldown reduction
more poke & combos

Marks: Magic penetration
more damage output

Seals & Glyphs: Scaling armor and magic resist
more survivability late game

21 - 9 - 0

21 offense
most damage output

9 defense
survivability

0 utility
no mana

Early game:

Fiendish Kodex - cooldown reduction for more poke and better farm
Hextech Revolver - spellvamp for sustain
Sorcerer's Shoes - first magic penetration to pierce through the enemies first magic resistances

Upgrade for Will of the Ancients last, however earlier, if spellvamp is needed

3 Core Items:

Wooglet's Witchcap
The Main Core AP item:
Most expensive of all, rush first, if enemies mainly physical damage
boosts your ability power value percentually, gives armor and Zhonya's active (stasis)

Luden's Echo
take first, if you need more aoe (splash) damage, good against grouping enemies
boosts your movement speed

Rylai's Crystal Scepter
get first, if more health is needed and enemies not primarily physical damage
adds a nice slow to all your spells, good against fleeing enemies

Late game:

Upgrade now Will of the Ancients (, if not done already)

Void Staff - gives more damage, because of percentual magic penetration

Defensive alternatives:

Abyssal Scepter - gives magic resistance, only useful, if enemies do much magical damage. However Void Staff more viable regarding damage output as soon as enemies magic resist is over 53.7!

Monflair Spellblade - gives still some ability power, more armor and magic resist as well as tenacity, if much CC

Banshee's Veil - gives magic resist and also health, very defensive item,
useful passive spell shield to counter CC

Upgrade Sorcerer's Shoes to alacrity for further movement speed

1st: Q "Bouncing Blades"
best for poke and farm in early game

2nd: W "Sinister Steel"
most damage to enemy team, because aoe damage and spammable due to low cooldown

3rd: E "Shunpo"
single target spell, that's why at last

R Ultimate: "Death Lotus"
Whenever you can, since it does most multiple target damage of your skills

Flash: best for engage, escape & chase
Barrier: more survivability during combo & ult in team fights

Alternatives

Ignite: more kill potential, only 1 target
Heal: more team utility, multiple ally targets, slower & less defensive than barrier

Pros:
- OP in this mode after some kills
- more survivability due to some defense and sustain

Cons:
- mostly banned because OP
- can still be shut down easily by CC