Hello, my name would be Mr Acurite, and I'm a low-silver jungle main, but I also enjoy the top lane. I would like to begin this all by saying that I'm not professional by any means, but this guide is not entirely intended to teach how to pull off jungle Ryze, but it is for the purpose of bringing attention to the fact that it can be pulled off. Right now, I'm sitting on a 75% win rate with him, in about half of which I carried to some extent. Now, to the guide.

Pros:
-Fast clear after first back
-High burst damage
-Decent bulk
-Good utility
-Decent split-pusher
-Can provide magic damage to an otherwise AD laden team
-Skullf***s the meta, other teams can't predict what you might do
-Moderate carrying power
-Can counter some other junglers

Cons:
-Horrendous first clear
-Extremely blue buff dependent early game
-Can get very low on jungle clears if you don't kite the buffs
-Ganks somewhat difficult depending on skill with the champion
-Can only solo dragon at level 11 or so

With Runes and Masteries, Ryze benefits significantly more from mana and mana regen than other champions. This remains true in the jungle, perhaps more so due to the requirement of tons of mana in order to have a good clear time, as even at level two through four/five, your mana is still being used faster than blue buff can provide it, even with just your Q. Spell vamp quints are extremely useful, as the massive ability spam Ryze is known for is going to do the vast majority of your damage the entire game, and healing 6% of that is going to add up, fast. Flat armor seals are standard, helping you survive your early jungle clears, and still aid late game by increasing bulk, keeping in mind that Ryze makes an excellent off-tank, so you will end up tanking some tower shots or enemy fire. Glyphs I split between flat mana and flat mana regen, with the base mana aiding not only in ability spam, but also increasing damage as Ryze scales with it, along with AP. Marks can be up to your discretion, with scaling mana being my pick, as it gives ~190 mana at level 18, and surpassing the flat mana marks by level 6, with the base mana once again massively aiding spam and damage. For masteries, I'm currently running 0-9-21. Defense has 2 points in recovery, and enchanted armor. 1 in tough skin and juggernaut, and 3 in veteran's scars. This gives a little bit of bulk, and aids jungle sustain. Utility has 1 point in phasewalker, strength of spirit, alchemist, runic affinity, culinary master, wealth, and wanderer. It has 2 points in greed, and 3 points in fleet of foot, meditation, vampirism, and expanded mind. This gives even more jungle sustain, a little late game 'oomph' with the extra money, an extra potion early game, extra effective potions, and a little more ganking power with the extra move speed.

This section is going to be prefaced with a description of why we're building the items we're building, so that you can modify the build as you need to fit a specific game or team. The build that I go for is generally an off-tank sort of build, with the tankiness always mixed in because going full glass-cannon gets you killed very fast, and if you get ahead the last thing you want to do is throw a game through shut-down gold. The majority of the offense from this build is going to come from three core items: Muramana, Rod of Ages, and Frozen Heart. With this, you end the game at over 3k mana, and the Muramana toggled on adds multiple hundred damage to your ganks and other fights, occasionally turning the tide or nabbing an otherwise unlikely kill. This also affords you about 3k health, and the frozen heart means you end at about 200 armor on top of other items, so ADCs are going to have a tough time killing you, meaning that ganking bot lane is now easier. Now, the actual items I like to build are:
Start of game: Machete, 5 pots, ward trinket/sweeping lens
-Pretty standard, trinket is up to your discretion
First back (1075 gold+): Tear of the Goddess, Quill Coat
-Start stacking that tear ASAP, which is very easy while playing Ryze, and the Quill Coat is going to help a lot with mana sustain issues in the jungle
Second back, onwards: Catalyst of the Protector is nice, as well as the Glacial Shroud, but both should come after a pair of Boots of Mobility, or other boots. I prefer Mobility boots, but you might want extra speed or magic penetration, in those cases getting the respective boots.
Late Game: Because it helps get the other items, Spirit of the Ancient Golem should be bought earliest of your late-game items, which also synergizes well with the Rod of Ages. In some order, after those two items you want to buy the Frozen Heart and Muramana. The last items should reflect the game currently occurring, with Banshee's veil for some magic resist and health, Warmog's for further bulk, especially against AD with the armor, and going well with the SotAG, Will of the Ancients/Hextech Gunblade are also good choice for the spell vamp, and the Gunblade offers a 40% slow for chasing, as well as the further AD for pushing. The build also gives you almost 200 AD, making your (Muramana boosted) auto-attacks hurt more then you'd expect coming from a caster. Keep in mind, the SotAG, Frozen Heart, and your Q's passive already hit the 40% CDR cap, meaning that further reduction is wasted.

Pretty simple stuff: Q at 1, E at 2, W at 3, max Q-W-E, take points in R when you can. You max Q for damage, W for the utility and ganking power, and E for wave clear. Never try to conserve your ult, it's on an extremely low cooldown, as well as being very useful for clearing large minion waves and escapes. Don't use it to clear camps, unless you're certain you're good for the next 50 seconds or so at rank 1, 30 seconds at rank 2, or 10 seconds at rank 3.

For both farming your jungle, use this combo: Q-E-Q-W-Q-Auto attacks mixed with Qs-Q-E-Q-W-Q. This provides a very fast clear mid game, and is pretty simple to remember and pull off. For ganking, you can do a completely ridiculous combo that can often instagib a target if pulled off fully, but is hard to land in full, and is somewhat clunky mechanically:
Q-W-Q-E-Q-R-Q-W-Q-E-Q. The total mana cost is very high, but the damage can easily go into the thousands if pulled off correctly.
For clearing a big minion wave to push, ult, and then use your E to deal massive AOE damage, clearing the wave. Supplement with other ult-boosted spells if necessary.

This Ryze build is stupid, it's anti-meta, it requires skill and luck, and at least some pretty big balls. Good luck, have fun, and send any comments my way regarding how well you did, or rage at me for losing you a game. Either's fine.
Mr Acurite, signing off.