If you do not understand this game mode, watch this video: http://www.youtube.com/watch?v=VzxJzAgIS34

Katarina is a HIGH BURST damage champion with a decent amount of mobility and a passive which allows her to score multiple kills quickly and chase enemies with Shunpo, meaning you can kill enemies behind you without running backwards. Before I begin talking about her gameplay in this mode, I would like to point out a few things first:

-The only way you can kill an enemy before 15 minutes is by abilities which damage minions that happen to damage champions too. For example, Katarina can kill someone with her Q in the first 15:00 provided that it bounces off a minion first.

-Once you start lapping, you CANNOT walk backwards - this is why Shunpo is so useful, especially in scoring multiple kills (easy pentakill since the enemy team should be grouped)

Pros:
-Insane capability to zone and score kills early game
-Can score multi kills easily from base (flash + ward shunpo) if you died early
-You will shred enemies who pass you when running around the map
-Can ward shunpo to get past towers / fountain

Cons:
-Too Squishy to tank - you need someone to tank (especially towers) for you.
-Very difficult to pass fountain when you are the second last person 'alive'.
-Ult can be cancelled

During this time, you must be able to last hit as much as possible (at least 4 last hits per wave + cannon minion). Only use your Q when you're sure it will bounce off a minion and land on your enemy laner. Once you hit level two, Do the same but walk up to your opponent so that he / she and AT LEAST ONE MINION is within your W. Use Sinister Steel and run away with the provided speed boost. Once you hit level 4, you can do as above but instead of running over to the enemy (when they can expect your poke), shunpo to the minion closest to your enemy laner and use W.

Sooner or later your enemy will start moving away from the minions, although this is usually only after your first kill. To compensate for this, walk over to the enemy team's approaching minions before your minions get there, and as the enemy minions will follow you, you can lead them over to where your enemy laner is and proceed to poke them from then on.

Over time, you will become better at baiting your opponents into staying in lane (helps to keep them on > 100 hp then Qing, Eing to a minion and W'ing them for the kill). If you enemy leaves lane to go back or because they died, PUSH. If they recall more than twice, chances are their mid tower will go down and you will earn your team 100 gold plus more for yourself.

To make sure you don't accidentally auto-attack, turn the auto-attack setting off in the 'settings' menu under 'gameplay'. Also, do NOT ult!

If all the minions in the enemy's wave die, push and attack the tower.

Let your team's tank soak up all the damage from the turrets. Consider yourself a high priority player. If you take any damage at all, stop between the turrets and use your spellvamp on the approaching minions. When you pass around behind the nexus, be careful not to get executed by the fountain or nexus turrets. Once you survive the first round, get either a Giant's Belt if you have more than 1,000 gold, a Blasting Wand if you have more than 860 gold but less than 1,000 and Alacrity bonus to boots if you have less than 830 gold. Work your way towards a Rylai's Crystal Scepter. Always leave base with at least one stealth ward.

These should be your goals every lap:
-Don't take tower hits
-Shunpo across the fountain area
-Try to kill anybody who passes you

Games don't normally go on for longer than you can buy Rylai's Crystal Scepter but, if they do, consider getting Zhonya's Hourglass, Abyssal Scepter and other defensive AP items (Guardian Angel works too).

At this point, Put a ward in the circled area below so you can shunpo to it any time. Shunpoing, flashing and comboing is an effective way to kill people who may have just flashed away. If the passing enemies are tanky, consider a Morellomicon or a Liandry's Torment. Scoring a kill outside of the base means that you can continue shunpoing to score more kills and wipe out the enemy team. At this point, you can either save the Hextech Gunblade for its slow or sell it for money for a deathcap or a stronger AP item.

Katarina excels in defending due to her mobility and damage.
THIS GUIDE IS UNDER CONSTRUCTION. MORE COMING.