Not Updated For Current Season
This is how I play Udyr, my way. Udyr, through my eyes is a well built champion, despite his nerf. In the hands of a good player, Udyr can be a indestructible force. Opposite for a bad player. You need precise timing on when do use Turtle stance, gotta know when to chase and flee. And knowing when to use phoenix. Having those advantages, makes Udyr a well built champion.
I like Udyr with some Armor penetration. So buy Armor Penetration runes. I wont show runes in the rune slot, but I will tell you a basic set up for runes. With the Red rune, I like to go armor pen. With the Yellow Rune, I like attacks speed and critical chance. For the Blue Rune, Critical Dmg, or chance is good. For the Quints, u want 1 Armor pen, 1 CD reduction, and 1 Atk speed.
For the Masteries, I choose mostly Defense for a reason. IMO Udyr is squishy early game, and i hate that. Also who doesn't want more dodge? If you dodge you gain 10% run speed.
I start off getting boots of speed, just because I like grabbing Berserker's fast. Get turtle stance and time it right. Don't trigger it right away, use the element of surprise. Once your level 2-4, if your team mate is good, go for a kill! You're almost guaranteed one with Bear Stance.
You start grabbing some kills in this part of the game. Your chasing skills you be outstandingly fast, and you'll rarely die with a combo of Bear and Turtle. If you have been fed, even 4 kills, 0 deaths. You can solo 2v1, and 2v3 for a short while. You just constantly switch between Turtle, Phoenix, Bear, Repeat.