DPS Jarvan is one of the most effective ways to play him in my opinion. Why do i think this? I think so because i cut through tank Jarvans like a knife through butter. His abilities increase his DPS by such a ridiculous amount, that i fail to see why anyone would build him as a tank. He even increases his team's DPS through his flag and spear, providing a giant amount of Attack Speed and armor reduction. In fact, his armor reduction is so useful that i can even 1v1 tanks during late game. Why would i build him as a tank when he does so much damage?

As the royal family of Demacia for centuries, members of the Lightshield line have spent their lives waging war against any who opposed Demacian ethics. It is said that every Lightshield is born with anti-Noxian sentiment in his blood. Jarvan IV is no exception, even though he is the first Lightshield born to the age of League of Legends. As his forefathers had before him, he led Demacian troops into bloody engagements with Noxian forces, and on many occasions he has bled alongside wounded allies and fallen comrades. In his most crushing defeat, he was outmaneuvered and captured by a Noxian battalion under the command of Jericho Swain. This mistake nearly cost him his life at the hands of Urgot, but he was rescued by the Dauntless Vanguard, an elite Demacian strike force led by Jarvan's childhood companion, Garen.

Those close to him believed that his capture changed him. Xin Zhao was quoted as saying: "His eyes never seemed to look at you, only through you to something he could not look away from." One day, without warning, Jarvan IV handpicked a squad of Demacian soldiers and left Demacia, vowing to find "atonement". He began by tracking and hunting the most dangerous beasts and bandits he could find in northern Valoran, but he soon tired of such prey. Seeking something that only he understood, he ventured south of the Great Barrier. He wasn't heard from again for nearly two years. After many had assumed the worst, he returned to glorious fanfare on the streets of Demacia. His Demacian plates were adorned with the bones and scales of creatures unknown. His eyes bore the wisdom of someone twice his age. Of the twelve soldiers who had departed with him, only two returned. In a tone as cold and steady as steel, he swore to bring the enemies of Demacia to their knees.

"There is only one truth, and you will find it at the point of my lance."- the "last words" of Jarvan IV at his failed execution

Innate - This innate is just amazing! Who doesn't want 2.5 times a Madred's Bloodrazor on their first hit and every 6 seconds? Not that they last long enough for you to use it again...

Q - Initially, this skill is almost useless due to the enemy not having much armor and it being a percentage-based reduction. However, due to it being a percentage it scales amazingly well. By the time it's rank 5, it's more than half of Last Whisper! This is the skill that allows you to eat tanks alive.

W - This shield is actually really nice, but it's hard to make the full use of it outside of a team fight, making it pretty useless during early- and mid-game. Level this last.

E - This is the biggest early-game increase to your DPS and should be leveled first thing first. This skill is amazing, providing you with 52% attack speed when it is used. This is almost as much as a Phantom Dancer! It evens adds some armor and buffs your team. This skill is your bread and butter.

R - This is such an amazing ultimate! By the end game, this one skill will nuke the enemy for over 1000 damage before armor reduction. This little baby allows me to jump on a stupid Master Yi who stuck around too long, trap him, and kill him without losing even half my health. Level when available, and practice caution when using it! More on this later.

The Combo
The best skill combo to use is always:

Cataclysm
into
Demacian Standard
into
Dragon Strike
with a
Golden Aegis to survive a bit longer and slow them if Cataclysm breaks before they die.

Also, when escaping, chasing, or even initiating, use your E -> Q in order to knock them up and move around faster.

1st rule of Cataclysm: DO NOT JUMP INTO THEIR WHOLE TEAM BY YOURSELF. Their ranged WILL kill you through your ultimate, and once it breaks you're screwed.

The most appropriate use of this skill is during ganks and teamfights to make sure that speedy Master Yi or Teemo doesn't escape your wrath. SAVE THIS SKILL FOR THE FASTER CHAMPIONS OR ESCAPING CHAMPIONS. Do NOT use this against only a Kassadin or a LeBlanc as they can easily teleport out of it. Be careful of enemies with Flash as they can escape with it as well.

Enemies you SHOULD use your ultimate on:
anyone with a higher move speed than your team

Enemies you SHOULD NOT use your ultimate on:
Any champion with a blink, Flash off cooldown, or a skill that allows them to jump through impassable terrain.

When ganking, make sure that if you're after Katarina, Master Yi, Irelia or any champion who can blink to enemies that you tell your team to stand out of their range if any ally is on low health. When Cataclysm breaks, CC them as they WILL attempt a fancy escape if you don't kill them fast enough.

I usually take Flash and Ghost due to their great chasing and escape uses. With these, your ultimate, Youmuu's Ghostblade, and your W, enemies will NOT escape and you can escape ganks and losing teamfights with ease.

Also useful:

After this item build we are left with nearly max attack speed after your abilities and Youmuu's Ghostblade, a large amount of damage and armor reduction, and more than 3/4 of our attacks will hit for 260% damage. All of this, while we can chase, prevent escapes, and survive almost anything. "Nerf pl0x! Jarvan OP!!!!111!!! QQ," is what you'll be hearing from the enemy team as you topple their forces!