I believe many people will call me noob or at least unskilled about offering such guide to the community, but I think it still Is worth a try. I will probably see comments like "Omg, Ryze - jungle" or "wtf noob" but i don't care, as long as you go out there and try it out.

Never do this on solo/duo queue, because you are gonna get flamed by teammates and probably loose the game because of distraction. However it might prove a nice surprise to the professional ranked team you play against ;)

Greater Quintessence of Spell Vamp allows for easier jungling (it will be incredibly hard early lvls).

Mag. pen. from Greater Mark of Magic Penetration is just essential to Ryze as well as mana from Greater Glyph of Mana .

Mag. res. from Greater Seal of Magic Resist helps since you are not going to do any mag res items at all.

Spell vamp - helps ease the jungling a lot.

Cooldowns - essential for Ryze's combos and it also kicks off the endless need of Blue buff.

The Strength of Spirit mastery is incredible with Ryze and also helps early jungling.
Mag. pen. - essential again...

Rushing Hextech Revolver might prove a great tool for jungling.

Small boots for at least some mobility (it's not that expensive).

Tear of the Goddess - essential for ryze's dmg output and probably the best possible item for this hero.

Rod of Ages - The item that gives you the most survivability and in the same time dmg output for the least price. Also is incredible help for jungling because of it's lvl up heal buff.

Ionian Boots of Lucidity and Glacial Shroud are good for Ryze's combos and eases the need of blue buff all the time in combination with the masteries. The shroud also gives good temporary survivability for mid game with that armor.

Now you can either choose to build up Rabadon's Deathcap or Archangel's Staff depending on whether you have enough stacks on Tear of the Goddess or not. If you have like 300 mana bonus from the tear, better go for cap. But if above 700 mana, get archangel immediately.

Finish Frozen Heart and then go finish your spell vamp item. Both Will of the Ancients and Hextech Gunblade are good items... It depends on your team and the enemy team... If your team lacks cc, take gunblade. If your team has more Ap potential than Ad, take the will of the ancients...

However, if you don't feel like being able to cast enough spells to keep yourself alive with spell vamp. and you see that every time the same story repeats - You get stunned and you can not survive with spell vamp, then I recommend replacing the spell vamp item with a more solid survivability item like Banshee's Veil

It is a standard Ryze skill build. You can choose whether you'd max W or E first, depending on what your enemies do and what your team needs... However E is better for farming so I'd recommend it for players who like to stay longer on creep waves and camps...

Standard jungler spells...

Absolutely essential to start off with blue buff. In competitive play you will probably get hard invasion of your positions, that's why you will have to ask your whole team to be around and defend you or simply invade the enemy blue yourselves...

You should consider ganging on lvl 2 because this will probably be your only early/mid game opportunity to make your team some favor... That is because by rushing for hextech you will loose some early mobility and your enemies will always stand before you in boots...

However, that is not your primary concern. Your objective is to farm as much as possible before the teamfights start. So ask your teammates to slow down the push and play passively. You should have made your build at least to Glacial Shroud before the teamfights start. That way you will be able to survive them and deal tons of dmg at the same time...

A good hard pull will do you great job. As i said you can gang only on the pre-early game phase when people are only lvl 2 or less.

Late game try to keep up with the team. Lock the most mobile heroes. It is not a good idea to waste the prison for the dmg in a teamfight. Locking melee ad's or close range ap's and assassins is the best you can do... For example heroes like these should be your priority:
Ahri - She won't be able to use her ultimate for nothing. Neither deal dmg. and engage, nor escape.
Akali - Just like Ahri, she won't be able to use her main and most potent spell - Shadow Dance and will probably be forced to go in stealth mode.
Alistar - Will not butt kick anyone in your team for a while cuz huh, we all know you can't kill him for that time ;)

Heroes that SHOULD NOT be your priority are for example:
Karthus - Will keep spamming on your head anyways.
Anivia - Same reason as above.

Another thing about teamwork. DO NOT STRIKE THE LOWEST HP TARGET. Instead - use your ult to deal dmg to as much enemies as possible instead of wasting the splash dmg on the ad ranged who will probably be far away from his teammates. However if the ad carry is indeed CLOSE to his teammates, FOCUS HIM!

Finally, you will need to be extremely careful early game jungling because your blue buff is way too important for you. You can't afford to loose it, so ask your teammates to put up some scouting around the main entries of your jungle and watch if an enemy is missing from it's usual position... And hey! DO NOT BLAME ON YOUR TEAMMATES FOR BEING GANGED IN JUNGLE. In fact, a good jungler should be helping his teammates and say "ss" instead of them. There is not much to watch while moving between camps, instead of admiring the beauty of your hero, just look at the lanes and note down if something is not right. However do not get too distracted!

This build is incredibly risky, as it's name refers to... You can always get your game totally ruined by some jungle gang or invasion, or by simply FAILING the jungling or even by a teammate mistake...

However if your start is good, the play style can be incredibly strong and stable throughout the whole game and as you might all know, Ryze is a champion that does well with all kinds of enemies and occurs in all the hot areas unlike some heroes like Nunu who is incredibly good at bush fights but can be easily avoided and dodged on lane...

Late game, Ryze is as good as he would've been on mid lane, but as a jungler, you will now have a mid lane free for another mage. I advice you to have a mass mage taking Ryze's place on mid. That way you will have 2 good mages at late game who will deal a lot of magical dmg to both mass and single targets instead of only 1 mage for single targets...

Well, now you have a whole new vision of Ryze's potential. You might all have thought about this like that - "Oh hey, there is a gigantic wave of creeps there i should go farm it... Oh wait i have to waste ulty just so my teammates don't come and steal the rest of the creeps cuz i would've probably killed 2 with my Q and 1 with my W and max of 2 with my E and the remaining 10 creeps are free for my team to take cuz my spells are on cooldown". Here is the conclusion - RYZE IS NOT A MASS DMG DEALER. HE CANNOT OVERCOME A GOOD MID AP AT FARMING SIMPLY BECAUSE HE HAS NO MASS SPELLS AND CANNOT KILL ALL THE WAVE FAST ENOUGH... But jungle on the other hand is nice to farm with Ryze since there are max up to 4 creeps and is much easier to kill 3 enemies than to kill 7 enemies with 3 single target spells and it gives almost the same amount of gold. Farming in jungle with Ryze is more solid than farming on lane... And I say again - this is a reckless Ryze play style so expect to fail sometimes because of the reasons i mentioned earlier in the Pros/Cons section...

I really appreciate your comment Mintleaves and indeed it made me think about a lot of things in my guide. Now as I look at it, it catches the eye much more and attracts attention.

However I will not change my opinion about the early Hextech Revolver and my mastery set and I will explain myself. The masteries are like that for three reasons. Firstly I love the hp/regen from mana cap effect and it indeed is something that is underestimated. Secondly the more cooldown reduction i can get, the better. You know how Ryze's Q spell simply does not have cooldown if you have more than 35% cd. reduct. for at least 4 shots. And finally, the spell vamp is very important for the jungling. You get a total of 9% spell vamp only at level 1. Calculated right you get like 15 hp for each spell you cast. This allows for staying in the jungle for long enough to farm for hextech.

Here comes the whole story with rushing Hextech Revolver . When you get that item, you will simply be able to stay in jungle forever. In addition to your like 25% spell vamp you will have higher levels on your spells which calculated right will prove my point.

As you can see, I have changed the late game items, replacing the Hextech with Banshee because indeed spell vamp is just not as good as health and armor/mag resist for survivability.

I am sorry for putting up a replay so late, but I had other things to attend to. So here it is, a replay about Ryze jungling and in fact winning the game. http://www.leaguereplays.com/replays/match/2124089/