Ryze was reworked, and he can be utilized in the jungle by a skilled player. He offers high burst and good clear once given the mana to sustain his quality combos.

Start Hunter's Talisman and 3x Health Potion or Refillable Potion

The first clear is hard, so 3 potions is a safer start, if you are good at the clear, you can start refillable

Early Items - The cloth armor is optional, if you plan on making Zhonya's Hourglass or Iceborn Gauntlet, it makes the first few clears a bit less hurtful.

Core - Rod of Ages and Archangel's Staff are the old fall back. The new ryze does very well with cool down reduction, so CDR shoes are the usual, due to the absence on his normal items.

After testing, I realized that when chasing somebody, if you cast E, you stop moving for a second and this means your target might be out of range for the W. This is why I put in Rylai's Crystal Scepter as core, it offers the health, while also giving you more guarantee on the W if you can land an E.

Alt Boots- If you will end up wit max CDR without the boots, or are comfortable without it, then alternative boots can be used. Mercury's Treads against CC, Boots of Mobility for roaming, and Sorcerer's Shoes for damage.

AP Build- This lacks much CDR, but it will maximize your damage. You will be fragile but dangerous.

Defeinsive AP- This builds more CDR and defensive stats, while still offering the AP.

Other Items (These shouldn’t replace the core)

Morellonomicon - This works well with Ryze, offering AP, 20% CDR, and Mana, it can be used to replace the Rabadon's Deathcap or Void Staff in the AP Carry build

Lich Bane This can work if you are good at getting in the auto-attacks, the mana and CDR are helpful, and the Spellblade puts another big hit into Ryze’s combo and allows him to put bigger hits on structures. The early sheen can help with the clear both in terms of CDR and reliable damage that doesn’t cost additional mana before Ryze becomes powerful enough to annihilate the jungle.

Support Items - Ryze can take a supportive role with a few items, his E->W combo is good for catching people, since it works on people with speed boost like Volibear and even Master Yi. If you are fast enough Q->E->W->Q,

Iceborn Gauntlet can be good for kiting, the Armor also helps with the early clear.

Locket of the Iron Solari and Banner of Command are useful if your team doesn’t have someone with Aegis but the enemy team is high in AP, they also provide CDR.

Frozen Heart - If the team has two or three champs with attack-speed based damage, this can help in a team fight.

Banner of Command - This sports a bit of AP, but can also be useful in split push.

Defensive-

Guardian Angel and Zz'Rot Portal can come in handy, providing quality Armor/MR and utility to split push or dive if you have to.

Kiting is essential, you must be able to avoid a lot of damage early on in order to survive the first clear. This can be hard because W costs a lot of mana, but once you have blue buff, you can wait a bit before the early camps to regen enough mana to fight.

Gromp w/ Smite, wait for smite to come up, then blue with smite, then wolves, then scuttle if it is there. If you can get to raptors with smite, you can hopefully get enough for tear, which is more important than the jungle item, as Ryze really needs mana early on.

Because both E and W reset the cool down on Q, if you have the mana
Normal Clearing Combo, Q->E->Q->W->Q->E->Q

Scuttle combo, Q->W->Q->E->Q, to reduce the armor first and use as many Q resets as you can.

Raptor, try to mark a small one, then auto and kill with a Q to mark the other ones and have your next Q split between them. If you kill a marked target with Q, the mark spreads, if you hit a marked target with E again, it also spreads. It is way more expensive to spread it with E, because the mana cost of E increases, so try to spread it with last hits when i can. This also works for wolf and krugs. I ignore the little ones on the buff camps until the end.

Catch somebody- Q->E->W->Q->E->Q (Extend the root)

Run- E->W->Q or W->E->Q, to get the move speed from Q. This can be used to chase or run away. You can start with Q to try to slow them if you have Rylai's. These two combos also give you a shild for 3% bonus mana and 60% AP, which is pretty strong. You can use it for a quick trade to hopefully not lose too much health, as you can run away shielded, and even extend it with another E->Q if you don't feel threatened.


The normal clear combo can also be used if you are in a good position.

If you have a spell blade you should try to get in auto attacks during your combos, which is easier when you have landed the root.

The first rank has a very small range. You can use it to hop walls if you are chasing somebody, but the long cast time and graphical warning make teleporting into lane easily avoided by the enemy. You can use to to escape if you have the foresight in time.

The second level has a lot more range, and this can help you get closer to fights you are a bit too far from, as well as escape. Try to ultimate with team support, in a safe position or defensive position, as the enemy can easily collapse on your target location and kill you easily if you place the out-portal too aggressively.

Runes

Red: Magic Pen
Yellow: Armor (For early clear)
Blue: 6x Scaling CDR / 3x Mana Regen (Early Clearing Mana and CDR)
Quint: Move Speed (Ganks, chasing, positioning, kiting)

Mastery
I take the defensive side of the tree, because I am not the best at kiting, so I feel having some extra defense causes my early clears to be a bit safer.

Ryze is a farm heavy champ, so if you fall behind, you will likely not do very much damage. This is why I try to focus on farming early game until I can clear safely and reliably. Once I have tear+ runeglaive, i feel more comfortable in my ganks, it is much more risky to try and gank when you have low health and low mana during the early clears, but once you have some charges on Tear, runeglaive and shoes, your burst becomes a lot more real. If you are good at kiting, early ganks become more viable because you have much higher health, but mana is still a very limiting factor early on.

Ryze is great at making picks happen with the team if he can land the root, and also has good burst if he is safe enough to keep fighting. The long cool down on the root is why CDR is so valuable on Ryze, because otherwise he is very vulnerable when it is on cool down.

Support items like IBG can extend your combo potential by increasing the amount you can kite the opponent, especially if they are tanky.

I am not great, but Ryze has a lot of potential. He can be countered by any counter junglier that can take his farm or kill him and escape if nobody is willing to back Ryze up, but if he gets good farm or gets fed, he is a very dangerous carry with a good clear speed. His ultimate makes him able to have a bit more map pressure as a jungler, and his mana scaling and naturally high damage output allows him to carry some off-beat items with more success than others would have.