Udyr is a strong champion both in jungle and on solo top. He has one of the fastest jungle clear times ever and because he clears it so fast he could potentially have enough time to counter the enemy jungler. He is probably the strongest in a 1v1 against the enemy jungler also, the only champion who I could see outdamaging him in a head on fight is Lee Sin. Udyr is generally played as an initation or protection champion. Usually he is you will be in front of your carries to block skill shots and just generally be disruptive. His kit gives him a great ganking potential and also gives him a decent amount of AoE damage. In this guide I will be covering Jungle Udyr in as best detail as I can.

Pros

Good Mobility
High sustain
Fast jungle clearing
A lot of natural attack speed with passive and Q
Can replenish his own mana by damaging enemies with W active
Strong single target ganks with stun
Lots of AoE with R
Can hold lanes well against almost any type of laner after being in the jungle for most of the game.

Cons

Can lose mana quite quick without blue
Relies on snowballing early game to become a great threat late game
Slows cause it to be hard to close gaps with bear stance

Jungle Order

Wolves
Ancient Golem
Wraiths
Gank Mid
Lizard Elder
Mini Golems
Wraiths
Wolves
Fountain

Alternative Joungle Route
Wraiths
Lizard elder
Mini Golems
Wolves
Ancient Golem
Gank Top
Fountain

Counterpicks to Udyr are generally champions that can cc him for an extended period of time.
Some examples of counterpicks to Udyr are:
Anivia - This is because of her Q stun her ult slow and her wall. This allows her to kite Udyr very effectively
Trundle - His pillar of Filth allows him to slow Udyr effectively and, depending on the situation, potentially wall him off completely. Pillar of Filth also allows trundle to bring Udyr towards his own team and pushing Udyr away from his own and he can use his ult to focus him down.
Janna - She has a slow, a knock up and a knock back. This allows her to prevent Udyr from stunning anyone important in ganks.

Udyr works well with champions that have a greater range than him as he gives them a zone to kite with. His stun from his bear stance is great to protect the squishier members of the team.

The Marks are attack speed as this allows him to clear the jungle faster, his R on hit procs allow him to clear the jungle very quickly. These also allow him to replenish health and mana on the weaker camps by using his W
The Quints are movement speed as this allows him to get into the enemy jungle faster to counter or steal it. They also make ganks easier as you can close down the target easier by using his E.
The Armor seals make jungling easier as the creeps can't damage Udyr as much meaning, combined with his attack speed bonuses, he can potentially outheal it by using his W.
The Magic Resist Glyphs are there to make him less squishy in early level ganks high burst AP champions such as LeBlanc.

Flash is a must have spell as it allows you to potentially hop over a wall when the enemy is pursuing you, this can almost certainly allow you to make it away by using your E's movement speed.
Smite is just as vital as a jungler as it allows you to clear the jungle a lot faster, with a lower risk of dying on the larger camps, it also means you could steal a buff from the enemy jungler by waiting out of sight as they try to clear it. Clearing your jungle faster also means your half of the jungle respawns faster so you can out level the enemy jungler if they clear it slowly. Also smite is really useful on baron or dragon as it means that, if timed right, its impossible for the enemy to steal it without having smite themselves.

You should play Udyr as an initiator. When in team fights do as best you can to stun as many of the enemies as possible so your team can kill the enemy easier. You should also use your stun to defend your carry as effectively as possible. You should just generally be in front of your carry during the fight as it allows you to take the enemy skill shots instead of them.