This guide is not to let you know how I play udyr from the ground up. This is for my item build masteries and basic routes, altough with your build you will be EXTREMELY tanky as do a nice amount of damage output.

Pros

Very fast jungler
Very tanky
Nice damage
Fun to play

Cons

No gap closer

I max pheonix and turtle first getting a point in bear at 4. I wont level bear again until both turtle and pheonix are maxed. I get 3 points in tiger at the end and generally dont use it too much, the reason I prefer Pheonix to Tiger because AoE damage > Single target damage.

Udyr can run any runes, but I think the ones I run are the most powerful for him.

Greater Mark of Attack Speed - These help you clear the jungle faster, it also makes your Pheonix Stance proc faster.

Greater Seal of Armor - Armour is great for Udyr when he's being targeted, it makes him a little more tanky because I take a little less damage from AD champs with these runes.

Greater Glyph of Scaling Magic Resist - I run these Magic Resist runes because it gives my Magic Resist an extra push (The magic resist from my item build doesn't compare to the amount of armour).

Greater Quintessence of Movement Speed - Udyr is all about Nimbleness, so these runes get him around the jungle/map that little bit faster, it alsy helps you catch enemies with Bear Form.

I go 16/13/1 for my Masteries, I build into Quickness on the Utility tree for that extra movement speed, I build 13 into Defense for the extra dodge and armour, I also get 1 in Nimbleness so if I dodge I can move faster, this goes well with my seals.

Flash and smite are only summoners you should ever run.
Flash is just too op right now to pass up.
Flash bear stun is key.

But if you insist on not using Flash Ghost would be the only other option, it is highly recommended to use Flash though.

My jungling path is I start pheonix I pull wolves while i have someone from mid or top do a little damage to the wolves. Then I head to blue have someone pull blue to help me kill it. I have two choices at this point I either go to their wraiths if I think their jungler will be slow or I full clear my jungle by heading to wraiths, red, golems. After this I will keep clearing and just look for open ganks. Mid top are both very good to camp for your ganks.
So keep an eye on those two, to start your ganks run in from a brush (Hopefully they won't have Wards or you could have Oracle's), then ear stance to catch up to them and stun them, then use Pheonx Stance to deal some damage and just switch between Bear and Pheonix Stances, with help from a good teammates about 80% of your ganks will get your team a kill!

Starting Items

For my starting items I get Cloth Armour and 5x Health Potion, get this at the start of EVERY game, you will not be able to clear the jungle at early level without them.

Early Game

Boots - I build these into Mercury Treads later on.
Heart of Gold - This item gives me extra health, gold per 5 and it also builds into [[Randuin's Omen later on.
Mercury Treads - I get these boots because without them my magic resistance would be too low, the extra Tenacy is also nice if the enemy team has high amounts of CC.
Oracle's Elixir - This pot is so useful if there is an enemy Evelynn or Akali, etc. It also helps you spot Sight Wards and take them down.

Late Game

Wit's End - This item will help you proc your Pheonix Stance better, it will give you faster auto-attacks and give you 30 extra Magic resist (50 if it's fully procced).
Randuin's Omen - I love this item on Udyr, it gives you health, armour, hp5 and a passive of 5% cooldown reduction, it also has an amazing Active!
Trinity Force - This is a great item for Udyr, although it is overpriced it has everything Udyr needs!, health, mana, movement speed, attack damage and ability power, it also has a nice passive of a chance to slow an enemy.
Warmog's Armour - When you have this you will be an amazing tank, and most of the enemies will be scared to face you.
Atma's Impaler - This is an amazing item when you have Warmogs Armour, it will give you extra Attack Damage, crit and armour!

Finishing Items

Mercury's Treads
Trinity Force
Wit's End
Randuin's Omen
Warmog's Armor
Atma's Impaler

Your job for team fighting on udyr is to sit on squishies with bear stance phoenix while switching to turtle and flashing out once people decide to focus you like ******s. The more attention you draw as udyr in a fight the better, this means you are taking pressure off of your carries to do their job.
Also remember a flash bear stun on a squishy to start a fight can be clutch so keep your eye open for that.

Overall I think Udyr is one of the best jungler's and gankers in the game! His bear stance can easily secure kills for your team, Turtle Stance can also be a life saver but remember, you are not god, so don't play like your god.