In this guide I'm writing 2 item builds with the same runes masteries and summoners. This is essentially a guide for Katarina in blind pick dominion. The player thus choses said runes/masteries/spells, then decides which build to play during the loading screen.

Spellvamp Kat -
This build is the stronger of the 2 builds in general; however it is easily shut down. This is the build you want to use if your enemy has few or no CCs that can shut down Death Lotus.

AP Tank Kat -
This build is your fall back for teams that can shut down spellvamp Kat, and is good at bursting and soaking damage, giving teammates time to finish off foes. Enemies tend to like to CC and focus Kat, and this is a good counter to that.

Spellvamp Kat

Pros

• Good Sustained Damage with Death Lotus
• Good Burst Damage with Bouncing Blades and Shunpo
• Spellvamp keeps her health high, easily heals during and after battles
• Good Escape Mechanic with Shunpo
• Once fed can carry her team
• Once fed can 2v1 or 3v1 easily

Cons

• Easily shut down and killed with CC
• If not fed, loses a lot of power
• Moderate to low move speed late game

AP Tank Kat

Pros

• Decent Burst damage with Bouncing Blades + Shunpo
• Ability for Sustained damage if they waste their CC
• Can tank a good amount of damage then Shunpo away or to kill

Cons

• Cannot carry her team
• Cannot 2v1 as easily, if at all
• Must town or pick up health packs to heal
• Moderate to low move speed late game

The Good

Exhaust - This shuts down many ad champions, and provides Kat with a slow to keep enemies in her Death Lotus. My favourite spell ok Kat in any mode with any build.
Ignite - This spell gives Kat that little extra damage that could mean a kill and cooldowns refreshed. This is good for getting kills in the level 3/4 fight and shuts down champions that heal a lot (like Fiddlesticks, Swain, Vladimir) without having to level Preparation.
Flash - A powerful spell on any champion, this spell could mean the difference between chasing down an enemy with flash + Shunpo or flashing over a wall if there is nothing to Shunpo to or if shunpo is on cooldown. I often run flash on Kat on Summoner's Rift, but in Dominion the long cooldown (and lack of Majai's Soul Stealer) make it worse then Ignite.
Ghost - This, like flash, is a chase/run summoner spell which allows you to get to top faster at lvl 3, chase down running enemies or gtfo if needed. I find it worse then Flash since Kat can Shunpo or flash over walls to gtfo and flash is better at chasing short distances, while chasing long distances usually means you're dead after(and if) you get the kill.

The Bad

Heal - It's good for surprising your enemies by having extra health mid fight, but it's almost always inferior to exhaust mid fight and doesn't keep them in your Death Lotus.
Promote - This spell is meh... I don't really like it. It feels like a support's spell on a map that doesnt take kindly to most supports. Take it if you like it I guess.
Revive - This spell lets you get back to slaughtering your enemies that much faster. Since you take full defense you can get the mastery for it, but it still has a long cooldown.
Cleanse - Great you're no longer CCed and now you're going to do what? your Shunpo and BB are already on cooldown... maybe you can run? Not a very good spell on Kat.
Garrison - A good spell for defending or attacking turrets, but Kat has no CC that helps so they can just run away and wait till the garrison is over. Maybe it's useful for diving enemy turrets.
Rally - You're not going 21 in attack so you can't get the mastery for it... which means it gives you 10-35 attack damage. Exhaust with the mastery will increase your damage by more, and gives a great disable.

The Ugly

Smite - Ok you can counter heimerdinger kinda... better to just kill heimer then smite his turrets.
Clarity - I'm out of mana! Ohno!
Clairvoyance - Most of the map is already shown.. the rest of the map usually doesn't matter that much. Kat has no uber long ranged spells and isn't a support. Skip this spell.

Runes

For runes I basically take 10 magic pen and as much AP as possible. Ap/level pays off by about level 6, which is when you get Death Lotus, so that works out. I use the 1 hp and 1 flat ap quint to strengthen her early game to get or prevent the enemy from getting FB. These are more useful on Summoner's rift, but even in Dom a strong early game can be powerful.

Other good Runes

Armour or dodge yellows and magic resist blues are also ok on Kat, they make her more tanky which is always good, even with the spellvamp build. (The more work it is for the enemy to do x damage to you, the more x healing is worth).
Move speed quints are also viable.

Masteries

This build gets Kat Exhaust mastery for her exhaust, Archaic Knowledge if they start building magic resist, 4% bonus to ap, 4% reduced damage, 3% cooldown reduction, +hp, mr, armour, dodge, ap and flat damage reduction. Overall, it gives her bonuses to most of her useful stats and no mastery point is wasted.

Other good masteries

If you don't like Archaic Knowledge you can get the ignite mastery for a bonus to her early game. If you have dodge yellows you can get Nimbleness for the bonus move speed. If you really want heal or revive, take the mastery. Other than that there shouldn't be much variation on the build mentioned above.

Voracity - This is a great passive, it means every time someone you hit dies, you get Shunpo and Bouncing Blade off cooldown and Death Lotus's cooldown is reduced by 1/3. This means you can Shunpo to an enemy, kill them then Shunpo to safety, great for diving. Also gives bonus gold which is always good.

Bouncing Blades - A good damage spell, has a bad ap ratio and a good ad ratio. Use this to harass, half healing with Killer Instincts and increase the burst given by your Shunpo. Gives less damage per level then Shunpo, but increases the number of targets hit.

Preparation - If you activate this before Bouncing Blades, it grants full damage to each bounce and halves healing for a short time, good for healing champions or stopping potions. If you activate this before Shunpo you get % damage reduction, which is good in general. *Note - the passive bonus damage adds to her base damage and thus does not increase the damage of her skills, making it almost useless.

Shunpo - This is probably Kat's best skill. This skill simpily deals damage to 1 target and moves kat to them. This can be used on enemies or allies. Use this move to harass, last hit, damage enemy carries, escape, get somewhere faster, and get penta kills.

Death Lotus - This skill gives Kat sustained damage, and lots of it. If the enemy has CC that can stop death lotus (Stun, Fear, Silence, Knockback, Knockup) try to get them to use the CC before unleashing lotus and killing them. If you sense a team fight brewing, let the rest of your team go in first, then when you see most of the CC used up, shunpo to their carry and unleash lotus. When lotusing, pay attention to enemies health, if you can shunpo to them and kill them, do so - you stop lotusing, but shunpo does damage faster, and if you kill them you get shunpo back immediately, so you can bouncing blade and shunpo to the next target and kill them etc.
One final note about Death Lotus - it's Katarina's built in stealth detection system. Death Lotus will become usable if in range of any enemy champions, even stealthed enemies, and it damages stealthed champions too. If you see this light up with no enemies around, it means careful eve, twich shaco or teemo are nearby

Skill Sequence

Leveling

Basically R > E > Q > W, although you might want 1 level in W if you want another counter to healing enemies, or multiple levels in W if you like the %reduced damage it gives with E.

Harrassing

Against most enemy champions, the best harassing sequence is Q-E then run. Some enemies you'll want to W-Q-E or Q-W-E (the first to stop them from healing if they're using a potion or are Vladimir or Warwick, the second to reduce damage they'll do back to you like ranged carries).

Team Fights

Wait till CC has been used, then Q-W-E-R their carry. This is usually lethal, once they die, Q-E the next lowest health enemy, if they die repeat at will.

1v1

If you think you can kill an enemy 1v1, Q-W-E them, wait for them to waste cc then R until they’re low enough to E again and kill. Some enemies you’ll want to initiate with W-Q-E.

Spellvamp Kat

Start with Boots and 2 Amplifying Tome and take the middle capture point and town immediately. By this time you will have enough to get your Hextech Revolver, get it and run to help top as fast as possible.

The next item you want is to upgrade your boots. There are 2 kinds of good boots for Katarina, Sorcerer's Shoes and Ionian Boots of Lucidity. For Spellvamp Kat, Sorcerer’s Shoes are almost always better, however, if you expect to be building Void Staff later (maybe they have Galio), get the Ionian boots.

After that you want to upgrade your Hextech Revolver to Will of the Ancients and build a 2nd Hextech as fast as possible. This gives Kat 40% spell vamp so she will heal a ton during death lotus and shunpo. Once you have these you have her core build.
Either upgrades your 2nd Hexteck to a Hextech Gunblade or buy Rylai's Crystal Scepter next, Both give her a slow that helps chase enemies and keep them in Lotus. Gunblade increases her spell vamp to 45%, while crystal scepter gives her enough health to take the enemy’s burst and heal it back with her spell vamp.

After getting one or both of the items above, you’re going to want more magic pen. When deciding which item to get consider 2 factors – how much magic resist is the enemy building? How much magic damage does the enemy team do? If the enemy team is building a lot of MR but is mostly ad, you’re going to want a Void Staff, if your focus targets have less than about 70 MR or the enemy has a lot of magic damage, you’re probably going to want Abyssal Mask. If they have both high magic damage and high magic resist, chose 1 and stick with it.

Rabadon's Deathcap is Kat’s "time to roflstomp" item. It costs a lot and only gives Kat 1 useful stat – damage. However, it gives a lot of damage, especially if you already have some items that give AP. Get this item whenever you don’t feel you need one of the previously mentioned items more. Usually games in Dominion don’t last this long, even if you’re fed.

Tanky AP Kat

Start with upgraded boots and health pots if you want, your job is to rush top and help that fight, let someone without full boots go mid and bottom. As for which boots to get, Sorcerer's Shoes or Ionian Boots of Lucidity are best on her, the former allows her to do more damage with skills, the latter allow her to spam her skills more often. Get sorcs shoes if you expect to kill a lot of enemies or the other team won’t build much MR, get Ionian boots if you expect them to build MR or CC you a lot. (The faster your shunpo cools down the faster you can get out of there when the going gets rough.)

Next you want Rylai's Crystal Scepter for the health and the slow. Since you’re tanky, you’re going to need the slow if you want to chase someone until they’re dead, also the slow adds team utility since you can minislow a lot of people with Bouncing Blades and Death Lotus and megaslow 1 person with Shunpo.

Hextech Sweeper is a nice item on Kat because it gives health, cdr and ap, however it also gives a passive and active which are occasionally very useful, since Shaco, Akali, Evelynn, Teemo and Twitch are all more commonly found in dominion then summoner’s rift. If you’re against any of these, you might want to prioritise Hextech sweeper above Rylai’s.

Now that you have health, you need MR and Armour to go with them. Make a judgement call on if you need MR or Armour first, but usually I find the item I need next is Abyssal Mask. Basically from this point on buy items from the list below, unless you feel the need to get a Void Staff.

MR Items

Abyssal Mask – This item gives good ap, magic resists and an aura that is most effective against enemies with 40-80 magic resist.
Odyn's Veil – Another nice magic resist item that stores magic damage you take and gives you an extra 200-400 burst with a low cooldown. This item also gives hp so it synergises with a few other items that you might get next.
Force of Nature – The item with the highest MR in the game, grants you regen based on your max hp which should be 2.5 to 3k after getting sweeper and Ralyai’s and grants you bonus move speed. It’s a good item for making Kat tank, but doesn’t add any attack power.
Quicksilver Sash – Good Mr and cheap, with a proc ability that cleanses on a low cooldown. I’m not much of a fan of cleanse on Kat but this item is a must against some enemies such as Malzihar.

Armour Items

Thornmail – A cheap effective item that hard counters auto attackers. You have enough hp to make it effective, get it if they have 2 auto attack champs and another 1 or 2 physical champs.
Atma's Impaler – Gives ok armour and attack damage (aka Bouncing Blade and Death Lotus damage) – You have enough hp to make this item worth it, but it’s not that great. The crit is wasted on Kat. This item would be good if they have a medium amount of AD and you already have Odyn’s Veil or another item that gives more hp.
Frozen Heart – Gives further CDR which will be wasted if you have Ionian boots and Hexteck Sweeper. However if not, this item gives amazing armour, a good anti-auto attacker aura and CDR. This item is better than thornmail but costs a lot more.
Zhonya's Hourglass – Gives ok armour and good ap. The active doesn’t synergise with Kat’s skills well (imagine death lotus continuing while you’re invulnerable :) But it’s still a good active on any mage and counters a few champs like Karthus.
Guardian Angel – This item isn’t as good on dominion as Summoner’s rift, but it still gives good armour and ok MR in 1 item. Meanwhile, who doesn’t want to be revived without giving up any gold once every while?

I hope you liked the build, leave comments please, go forth and get loads of triple kills, maybe a few quads and a penta or 2.