The thing I enjoy about playing as Udyr is that he's one of the few champions with many different build paths to explore depending on your play style. That being said, I play him a little different from the others above me.
Personally, I max Tiger before maxing Phoenix simply because of the insane amount of damage you can put out in a single attack. It makes ganking a vulnerable lane all the more satisfying. Also, I max Turtle last (leaving Bear at Lv3) so that I can have that strong dueling potential in the beginning of the game as well as becoming a strong tank in the late game.
Runes: Flat AD Marks/Quints, Flat Armor Seals, MR/lvl Glyphs
Masteries: 0 - 27 - 3 (The 3 in utility is for the Wanderer mastery)
Start of the Game:
Buy: Machette + 5 HP Potions
Lv4 Jungle Path: Wolves -> Blue -> Wraiths -> Wolves -> Red -> Wraiths -> Look for a vulnerable lane or continue jungling.
Early Skill Order: Lv1-Phoenix, Lv2-Tiger, Lv3-Tiger, Lv4-Bear
Phoenix is the best skill for your initial clear of the Jungle, always take it first. When you get Lv2, take Tiger and head to the next camp. When you arrive, activate Phoenix and immediately use Tiger on the big minion of the camp. The added AD you get from Phoenix powers up Tiger so that it will do more damage, thus taking out the big minion quickly enough so you don't take a large amount of damage. Then switch back to Phoenix so you can deal some extra AoE damage to clear the camp. Continue clearing camps until you reach Lv4 and get Bear Stance. At this point, your movement speed will still be relatively low even with the active of Bear, so unless you see a really overextended enemy, you might want to just continue jungling. When you get 850 gold, go home and buy Boots of Speed and a Spirit Stone.
Core Buys: Boots of Mobility, Spirit of the Ancient Golem, Aegis of the Legion (Gold Value: 5550)
Skill Focus: Q>E(Lv3)>R>W
This is the part of the game where you really have to watch everyone's lanes while you travel from camp to camp. The laning phase is one of the most vital parts of the game, and if you can't reach your allies when in danger, then you're pretty much useless. Although my main focus is Tiger, leveling Bear periodically when Tiger isn't available allows you to reach your enemy faster before they can run away in fear. Every little bit of movement speed helps Udyr considering his biggest weakness is his lack of an immediate gap closer. Also, the best part I've found about taking Tiger before Phoenix is the amount of damage you can deal and still be tanky. You can build nothing but tank items all game long and Tiger's active will do significant damage to just about anyone.
Early to Mid Game Ganking:
Another thing I enjoy about Tiger stance. It's the only stance of Udyr that can be "loaded" and combined with his other stances and still be relevant when you reach your enemy. Like all of Udyr's stances, they have an active ability that remains even when you switch stances for a short while. Tiger is no different, except that you get the full effect of it if you reach your opponent in time. As soon as you decide to engage, activate Tiger and shortly there after, activate Bear to chase down your victim. When they get hit by your first attack, not only are they stunned by Bear but they took the full effect of Tiger's active as well! Continue the assault by switching into Phoenix and running alongside your fleeing enemy, letting the passive flames do their job. If you stop to attack, you may not be able to attack them again. As soon as Tiger is off cooldown again, activate it and hit them with a Phoenix empowered Tiger strike for even more damage! With your lane ally's help, you should net an easy kill.
Make sure to periodically buy wards and place them in your opponent Blue and Red buff areas. If you notice the enemy jungle elsewhere on the map, sneak into their jungle and try to snipe a buff away from them. Constantly switching between Tiger and Phoenix makes quick work of the buff monster and your Boots of Mobility + Bear makes sure you get out of the enemy jungle as quickly as possible.
Situational Buys: Runic Bulwark, Randuin's Omen, Frozen Mallet, Iceborn Gauntlet, Trinity Force, Warmog's Armor, Frozen Heart, Spirit Visage, Blade of the Ruined King (So you can solo Baron xD ).
All things said, you're a tank built around the enemy damage dealers. Choose your final items depending on who is the most fed on the enemy team or if they're running multiple AP champs. When the fight happens, and you can't sustain the initial wave of attacks, you're useless. With no way to immediately close the gap in a team fight, you'll have to rely on your allies to poke them down as you dance around at the towers. However, when that big fight happens, make it your job to sprint toward the most vulnerable carry of your choice and simply bully them out of the fight so your team can clean up the rest. Your high HP/Armor/MR and natural sustain through Turtle stance should buy your team enough time to whittle their team down into a retreat. Also, Bear Stance's stun is useful versus champions who have channeled ultimate attacks such as Katarina, Galio, Miss Fortune, or Nunu just to name a few. A quick stun will render their most powerful ability useless, just be sure to save it for after they use the ability and not before. You're also very good at picking off stragglers on the enemy team. I do suggest you add either a Frozen Mallet or Iceborn Gauntlet to your build, but not both. Either way, when you catch up to them, they're not getting away and your team can capitalize on a crippled enemy team.
TL;DR: I enjoy using Tiger early on over using Phoenix early on because I think the lane ganking potential is much higher.