Udyr is one of the stronger jungles in the game, you may choose to play him more damage or in a more tanky variety in which I enjoy to play. My reasoning for tiger over phoenix is that tiger does more singe target damage, you only really need one point in phoenix to jungle really fast, and in my opinion tiger and phoenix should be at level 5, bear should be 3 because it only increases move speed, which I have spec'd a lot into, phoenix to tiger stance is also a very devastating combination, although i do max phoenix last because i focus on single target until later in game.
I choose the flat attack damage marks for the increased early damage, although others that could work are armor penetration and flat attack speed, personal preference.
I choose flat armor seals for jungling because all camps do physical damage, however flat health seals may also work.
I use stacking magic resistance runes for later game when you have to worry about magic damage, I cannot think of any other glyph that would fit here.
I personally use Move speed quintessences because i like to go fast, which does increase jungle speed, other mentions are flat ad, armor penetration or attack speed, maybe even flat health.
i choose to spec into defense because i play the more tanky udyr and rely on his base damages which are decently high. i use a few points in utility so i can go just that little bit faster to catch an early gank. the points i take first are resistance and hardiness for added durability along with basically the whole tree.
You may choose a basic 21/9/0 set for greater damage, or a less common 15/15/0 for equal amounts
First you must be thinking "Why no wriggles?" well you can build it, but in this guide I take Phoenix stance first, this gives you a way faster clear time, but If you are falling behind I would suggest wriggles lantern or if you don't have the runes you can go cloth armor into a wriggles for the starting junglers.
I always start with boots and three health potions, but some may still choose to use cloth armor and five health potions, the final boots you get are always based off of your enemy team, if they have heavy ad or auto attacks, you should take ninja tabi, if they have lots of crowd control or AP I suggest mercury treads.
Be sure that if you get the heart of gold you get it BEFORE 15 minutes in, otherwise it will be to little to late. Next I always get my trusty Wit's End, it does a lot even if the enemy team doesn't have a lot of magic damage, because you hit really fast as Udyr and that extra 42 magic damage just adds more they have to build to fight you.
Trinity force at the end of build is also optional, but not situational, there's no reason to not get it other then preference, its a great item for someone who toggles abilities like they are on fire
Basically from here your last 4 items will be different every time. If the enemy has lots of magic damage banshees veil, quick silver for that pesky Malzahar and Warwick
Force of Nature can work too. Thornmail, Randuins Omen and frozen heart are also good items for the heavy AD enemies,
If really desired you can build some more health, although not entirely necessary, you can get Warmog's Armor, Frozen Mallet and even Atmas Impaler for that damage boost.
If you really want it then go for it, it is a great all around item, but if you died once you honestly will probably die the second time too.
If you like to do damage, and don't like tanky Udyr you may want to consider building a wriggles lantern instead of heart of gold, or getting a madrids blood razor, a blood thirster with frozen mallet warmogs and atmas is a great combo for tonnes of damage.
I know you're thinking "It says Tiger Udyr but you took phoenix stance." but thats simply because its the fastest way to clear the jungle early. I take my first point in phoenix stance, second in bear to gank 3rd in tiger for damage and 4th for turtle for sustain so i can give the next blue buff to my middle lane, from here i put as many points as i can in tiger stance, putting points in turtle whenever tiger is unavailable. next you max turtle then phoenix. Again you're thinking "Why phoenix stance?" well its simple, going from a bear rush in to stun to phoenix stance (increasing attack damage by 24) then tiger stance for more attacks speed and even more damage will give udyr his highest damage potential. then i put bear stance to level 3 with my final points.
I choose Smite when jungling for an obvious reason, it makes it faster easier, and allows you to counter jungle. It is simply a must when jungling, being able to without smite does not in fact make you a better jungler.
I chose flash for the simple fact: Flash bear stance is scarey. it also gives you an extra escape when needed or allows you to get in there for that hit or follow up their flash.
Ghost: A great chase spell, and good for running away too, bear stance ghost scarey, but not as surprising as flash bear.
exhaust: Again great for gap closing and good with bear, but not as surprising, but if you spec into the offense tree a little you can use it for the armor and magic resistance debuff to do more damage
Heal: everyone hates when the enemy heals, ALWAYS.
Pros / Cons
-survivability AND shield
-self damage buff
tanky and lots of damage
-Drops fairly fast when focused by 5, (don't know if pro that tank is focused or not)
-requires practice to learn his combo's
-ganking requires proper knowledge of when and where
Creeping / Jungling
When jungling Udyr, you should start at wolves then go to blue buff, from here you have a few choices, gank top, bot or mid, in which case get bear stance. (whichever your blue is closest to and is pushed out past the river you may have to come through their tri bush to successfully gank) OR you can go to your opponents red buff and take it it, most jungles are slower then udyr, but if your opponent has a jungler that starts red there is no point in wasting time going to it when it won't be there, other junglers that are slower (so not mundo shyvanna udyr, all i can think of) you can ambush at their buff, you may want to gank then go and wait because the enemy jungle tends to be fairly slow compared to an Udyr with phoenix stance. otherwise you can go to your wraiths then red buff in this case take turtle, then gank again if there's an opening or golems. repeat as necessary, you may also just go into your enemies jugnle and take random creep camps that they haven't got yet, HOWEVER when you go and counter jugnle like this, be sure to always leave one of the little minions behind so the enemy's camp doesn't respwan for them, if you control the jungle with your teams influence however you may choose to take all of it so you can get it when it respawns. Just be sure whe counter jugnling you watch for lanes to go missing that may come to fight you and the enemy jungler could always be there, and make sure your own team mates know you're in the enemy jungle and follow if their lane leaves to come to you.
*Note* use turtle stance often in the jungle in order to regain mana.
When fighting, be sure to toggle your tiger and turtle stance very often.
To initiate be sure to land a bear punch, then phoenix if you know they wont get away, and swap to tiger and proceed to kill them.
when in the jungle use your skills: phoenix then turtle then tiger, this will ensure high damage dealt and low damage taken
When fleeing enemies, toggle your bear and and turtle stances as often as possible, be sure to not oom (out of mana) yourself and die.
This is how i play Udyr, its also my first complete guide since I've learned how to properly play. enjoy and I would enjoy feedback.