Karthus jungle is a legitimate, situational pick for when their team is unlikely to counterjungle you due to playing someone like Amumu and you need some more AP on your team.

Pros:
Amazing lategame
Can't get shutdown in lane

Cons:
Not as much farm as lane karthus
Easy to counterjungle

Acronyms used:

AD = Attack/Physical Damage
AP = Ability Power or Magic Damage
HP = Health Points
CC = Crowd Control
MR = Magic Resistance
AOE = Area of Effect

Changelog
July 23 2013 - Added in Faerie Charm vs. Machete in Item Builds
July 22 2013 - Added in masteries section.
July 21 2013 - Added in ganking and runes sections.

Flash is taken because Karthus has no escape. Although not all Karthus players need an escape, Flash is useful for most players who need a quick escape or gapcloser. Of course as a jungler, a Flash over a wall could let you Smite steal an objective. These are things to consider if you choose not to take Flash.

However, if you feel extremely confident that you will not need an escape, gapcloser, or blink over a wall, Exhaust can be taken for more CC in ganks, more presence in a teamfight, and makes it easier to land Lay Waste. The Exhaust used in a gank pretty much ensures a kill and in a teamfight, Exhaust can peel for your carries and be used aggressively with much better results than Flash, since Exhaust slows the target for your allies, while Flash brings you closer to them which only benefits you.

Teleport is also an option because more global presence is exactly what Karthus needs. Teleport allows you to get in a quick gank when you see people out of position. It also lets you recall when low from taking poke from Lux or Nidalee then get back into any fight that may occur immediately. Because of your waveclear with Defile, Teleport could also be used to split push if you choose to do so.

Greater Mark of Magic Penetration is standard on all AP carries. As a carry, its your job to deal damage, so in the marks category, that pretty much leaves Greater Mark of Magic Penetration and Greater Mark of Ability Power.
All champions start with 30 MR, reducing your damage by 23%. However, running 9 Greater Mark of Magic Penetration lets you ignore 7.8 MR at the expensive of the 5.3 AP from Greater Mark of Ability Power. With 22.2 MR, only 18% of your damage is ignored. Although the Greater Mark of Ability Power does provide more damage in the very early game, the more damage you build, the more is negated by magic resist. So as the game goes on, its clear the Greater Mark of Magic Penetration outscales the Greater Mark of Ability Power.
As an example, with Karthus' most spammable skill, Lay Waste, you'll deal 40 + 0.3AP magic damage per Lay Waste, doubled when hitting an isolated target.
(Note all champions have 30 MR at level 1, assume that the target is not isolated)
At level 1 with Greater Mark of Ability Power, Lay Waste will deal 40 + 0.3(21.8) damage before MR, which is 46.5. After the 30 MR, Karthus will only deal 35.8 damage.
At level 1 with Greater Mark of Magic Penetration, Lay Waste will deal 40 + 0.3(16.5) damage before MR, which is 45.0. After the 30 MR - 7.8 Pen = 22.2 MR, Karthus will deal 36.8.
So even though Greater Mark of Magic Penetration outscales Greater Mark of Ability Power lategame, it even outscales it at level 1, making Greater Mark of Magic Penetration the clear choice.

Greater Seal of Armor is chosen because there aren't really any other good seals. Greater Seal of Armor is great in particular because Karthus has an incredibly bad early game. The Greater Seal of Armor will allow him to survive attacks from jungle creeps and any auto attacks from ganks Karthus chooses to make or invades. With Greater Seal of Armor, Karthus will take an additional 12% reduced damage from all the above sources.

Karthus scales incredibly well into late game and 9 Greater Glyph of Scaling Ability Power will give him nearly 600 gold of free AP (27) once he hits level 18. Some people might disagree with me, but I choose not to use Greater Glyph of Magic Penetration because magic pen is built to counter MR. So I think that magic pen would have the same gold value as MR. If MR is worth 20g/MR, then 9 Greater Glyph of Magic Penetration, giving 7.5 magic pen would only be worth 150 gold.

Greater Quintessence of Ability Power is chosen over Greater Quintessence of Magic Penetration for the same reason as why I chose Greater Glyph of Scaling Ability Power over Greater Glyph of Magic Penetration. I decided to choose Greater Quintessence of Ability Power over Greater Quintessence of Scaling Ability Power because the Greater Quintessence of Scaling Ability Power will only exceed the the Greater Quintessence of Ability Power after level 12. After that point, I feel that any of the stats offered by the quints are not too useful, and the flat 14.85 AP from the Greater Quintessence of Ability Power would help Karthus much more in his early game than the Greater Quintessence of Scaling Ability Power would in his late game.

Note about magic pen vs. AP runes
Many people are probably wondering why I choose AP over magic pen in my glyphs and quints but not marks. The reason is that AP marks simple don't give enough AP like the Greater Glyph of Scaling Ability Power and Greater Quintessence of Ability Power do. If the Greater Mark of Ability Power gave enough AP to outdamage the Greater Mark of Magic Penetration early game (like pre-6), then I would've chosen them instead.

If you're taking Exhaust instead of Flash, move a mastery point from Havoc into Summoner's Wrath .

Sorcery is taken since, in the first tier, there's nothing else useful. Karthus generally won't take the summoner spells for Summoner's Wrath and being an AP carry, he doesn't need attack speed. However, being a jungler, he does need Butcher to clear camps faster.
Blast gives AP per per level, which is much more useful on an AP carry than bonus AD from Deadliness since they don't autoattack much and Destruction since they typically don't push towers on their own either.
In tier 3, Arcane Knowledge and Havoc are the only choices. A bonus 0.67% damage per rank on Havoc doesn't make that much difference, so only 1 point (the remaining point after maxing all useful masteries) is put into it.
AP Carries are not autoattack reliant so lethality and Brute Force are much worse options than Mental Force and Spellsword which provide Karthus with more AP to deal damage with his Lay Waste, Defile, and Requiem. Spellsword provides Karthus with a small amount of damage on his autoattacks when he builds AP, which you can be sure that he'll be building a lot of.
Frenzy is not an option in tier 5, and being primarily magic damage, Karthus does not need Sunder and would benefit much more from Archmage .
The last point is put into Executioner for the bonus damage, which works especially well with Karthus since he'll be sniping a lot of low HP targets with Requiem.

Karthus has a horribly squishy early game, which is why I decided to take points into the defensive tree instead of utility.
Karthus doesn't use need the bonus gold from Summoner's Resolve and the HP regen from Perseverance is not noticeable enough to run it over Durability .
Tough Skin allows Karthus to tank the jungle creeps better, adding on more survivability in the jungle which Karthus desperately needs due to having no innate sustain skills like Bountiful Harvest or Soul Eater.
Hardiness is chosen over Resistance since MR is only useful against champions and armor will allow Karthus to survive more attacks from jungle creeps and any auto attacks from ganks Karthus chooses to make or invades. Bladed Armor isn't chosen because Karthus needs survivability, not more damage in the early game. Besides, I'm scared of running Bladed Armor because I know that one day, it'll mess me up if I ever want to give away my buffs.
Veteran's Scars is the best option here, as a bonus 30 HP is much more useful in the jungle than 1 Unyielding , Relentless , and Safeguard.

Tear of the Goddess gives you a mana pool so you can spam your Lay Waste and Defile. Don't forget to use your Lay Waste whenever you can while you have blue buff to keep on stacking your Tear of the Goddess. To stack it even faster, leave your Defile on when in fountain and recalling.
It also builds into Seraph's Embrace to give insane damage. Seraph's Embrace will provide Karthus with 131 AP at level 18, along with a shield giving an extra ~550 HP when used at maximum mana.

Rod of Ages is purchased because Catalyst of Aeons is a great item to have early on, and it gives health so you don't just get bursted easily, mana to use Lay Waste and Defile more, while synergizing with Seraph's Embrace, the 850 mana from Rod of Ages provides Karthus with an extra 19.5 AP.

You should get your Tear of the Goddess and Rod of Ages ASAP, so you can get them stacked as soon as possible. You should have them by the 20 minute mark, take taxing from your lanes if necessary to get these items, just don't take too much.

Zhonya's Hourglass is overpowered, it gives amazing defense, with the armor and active, tons of AP, and builds out of Seeker's Armguard which should be built early when playing against a heavy AD team, this is especially important if you're being counterjungled constantly by someone such as Rengar. Using the active of Zhonya's Hourglass with Defile on when you get caught can turn a bad situation into an ace for your team.

Rabadon's Deathcap gives Karthus more damage, that's all there is to say about this item. Other than that, the only thing to mention is that the later you build the item, the more efficient it is.

Void Staff lets Karthus deal damage even if the other team stacks MR. Void Staff IS A NECESSITY IF YOU WANT KARTHUS' DAMAGE TO STAY RELEVANT LATEGAME OTHERWISE THE OTHER TEAM WILL JUST STACK MR AND YOU'LL DO NO DAMAGE.

What about Liandry's Torment and rylai's?

I have considered these items but, your Wall of Pain already offers an AOE slow, that gives the same effect that the Liandry's Torment magic pen would. However, these 2 items take up valuable item slots from Karthus and in addition, for it to be most effective, BOTH these items would have to built. When missing either of these items, the combo is much less effective.
I also feel that most of Karthus' damage should come from his Lay Waste, to get a good amount of CC and burn off from your Defile, you'd need to be in close proximity to a lot of people. That's a situation less than desirable for Karthus. I feel that Defile should be used as a "I'm going to die anyways so take this damage".

Why no Hunter's Machete?

Fast junglers like Shyvana and Karthus do not need a Hunter's Machete to clear faster. In addition to that, Karthus can't build any items out of the Hunter's Machete. Spirit of the Spectral Wraith is a subpar item for Karthus, unless you're planning to not be an AP Carry and just bumrush objectives by stacking spellvamp. Spirit of the Elder Lizard isn't a good choice for obvious reasons, and Spirit of the Ancient Golem doesn't give the stats needed for an AP Carry to deal damage.

As most people are aware, Karthus will run out of mana even with blue buff because of Lay Waste spam and the upkeep cost of Defile, because of this, I chose to start a Faerie Charm and Mana Potions on Karthus. The main advantage of this start over Hunter's Machete and 5 Health Potions is that with a Hunter's Machete, you're getting 120 gold back when you sell it, meaning you'll need 580 gold to build your Tear of the Goddess, starting with a Faerie Charm means you'll need 520 gold in comparison, and you'll also get the extra mana sustain through the Mana Potions.

Lay Waste is your main damage tool, so level that up first. It deals more damage if you only hit one enemy with it, so try your best to hit an isolated target. This is your main damage tool, hitting your target with your Wall of Pain makes it much easier to hit them. When playing against Karthus, always try to walk into minions when being hit by Lay Waste so you'll always take less damage.

Wall of Pain is Karthus' CC tool, it's great for slowing down a target in a gank, or dissuading the enemy team from walk in a certain path. However, Karthus does not really need the CC so this is maxed last.

Defile is a large AOE damage move. Keep in mind that with this, you can take down jungle bosses extremely fast. In the jungle, use Defile to kill the small monster so you can get the bonus damage by hitting the isolated large monster with Lay Waste. Toggling Defile on and off instead of just leaving it on 24/7 will save you mana, so when clearing the jungle or going for an objective, mash your E.

Requiem global ultimate, makes people die, then you get paid. Make sure you have coordination with your teammates so that you never miss a kill. Also, it is preferable to use this AFTER you die since while alive, you can be killed, silenced, etc. and that would cancel your ult. DO NOT USE THIS TO STEAL KILLS. USE IT TO:
Secure kills, lower the HP of the other team, putting them in a risky position should they choose to continue fighting your team or go for an objective.

I usually start blue buff, then wolves, wraiths, red buff.
However if the other team has someone like Nunu & Willump, Udyr, Lee Sin, Shyvana I'd expect to get counterjungled and move onto red buff immediately after I do my blue buff.
I would usually back once I get 520 gold so I could pick up my Tear of the Goddess.
Once you have blue buff, you should always spam your Lay Waste to get more stacks on the Tear of the Goddess.
ALWAYS WARD YOUR BUFFS if you are being counterjungled or THEIR BUFFS to counterjungle them.
If you see the other jungler in a lane far away from their buff, steal it if possible.
Always save your Smite after your first couple of clears. You need it to secure Baron and Dragon .

Use Requiem, get paid.

Seriously though:
These are the standard ganking paths that any jungler can take. Although its impossible to gank a properly warded lane, most lanes generally won't be properly warded and this map will help you circumvent wards. For example, coming from blue side, ganking top lane, going the path orange, purple to cyan or pink, you'll be able to circumvent all the wards in the river, but you'll still trigger the tribush ward. However, depending on how pushed the enemy laner is, you can probably kill him still.

Your Wall of Pain is like Shaco's Jack In The Box but easier to use. The idea of Wall of Pain is to cut off any escape route you think they might take. Of course they can always get around it, but proper usage of Wall of Pain will always slow down the laners. They'll always have to choose between walking around the wall and taking a ton of time doing that or getting slowed walking through the wall.

Use Exhaust on the enemy laner if you need more time to get the kill and you need to CC the target. Its extremely important to exhaust the enemy laner if you're ganking for champions like Fiora or Nidalee. They have no CC of their own so chances are, you'll have to exhaust the other laner to have enough to take them down, otherwise they're likely to just get away.

Karthus jungle is unconventional, and situtational, this is the most important thing to note. You cannot pick him for every team comp and he requires very specific situations to do well. Good luck and have fun.