Udyr is well known for being a Champion who is at the heart of Ganks. However, it is not heard of very often, that Udyr is also the main cause of many of the enemy champions deaths. I have practiced with this Champion for a while now, and have come up with what feels like the best Udyr build, at least for me. At the end of my last practice session against a friend playing Darius as a Melee-DPS Tank, I was able to not only turn his ~4000 HP into 0 HP, but did it only losing about 1/3 of my health, quickly regaining it by farming the minions next to me. At the end, I had built Executioner's Calling, Berderker's Greaves, Thornmail, Phantom Dancer, The Bloodthirster and The Lightbringer.
Save Mana - Get Executioner's Calling
Like I said in the introductions, through trial and error, I got the build that I felt best suited myself and Udyr. So I compiled a list of what is needed for what. In the beginning, grab Brawler's Gloves, a Health Potion and a Mana Potion. During the first minute or two, farm minions, but try not to get involved with the enemy Champions, unless you and your laning partner (if any), are ready and good. Next, grab the Vampiric Scepter. It will help you regain health faster when using other stances, besides Turtle, and help you save Mana from constantly switching. My personal opinion would be to use Tiger Stance or Phoenix Stance to get Health up. After eating some more minions or some assists or kills (maybe all of them), go get your Executioner's Calling. That will complete your first build, help you stay in lane longer, and again, save Mana, from not switching to Turtle when low on Health.
Boots That Make My Fists Fly! - Berserker's Greaves
When you have the gold, and the time is right of course, return to the shop and get the Boots of Speed. You'll move faster, and no one will call you a noob for not grabbing shoes. Next is your Berserker's Greaves. Again, this way you won't be called a noob, and your Attack Speed is increased by 25%, complementing your Tiger Stance and Executioner's Calling combo.
Punching Me Is Like Petting a Porcupine - Thornmail
Right after getting your Berserker’s Greaves, grab Cloth Armor, or if you have the money, the Chain Vest. They will help protect you and will bring you so much closer to Thornmail. If you haven't grabbed the other ingredient, do so, and if you have the money, or when you do, grab the Thornmail. It will increase you armor greatly, and hurt enemies who are using basic attacks.
Soar Like a Phoenix, Claw Like a Tiger - Phantom Dancer
Next, grab a Dagger or two. Increases Attack Speed even further, and allows you to be all the more closer to the Phantom Dancer. After the Dagger(s), grab the Brawler's Gloves, and then Zeal. Zeal increases Attack Speed and Critical Hit Chance, as well as making you faster. After, if you haven't got a dagger, get it and the Cloak of Agility. After that, the Phantom Dancer is just a couple Champion Kills away.
I'm Literally Eating Your Face - The Bloodthirster
By this time, you're team has probably pushed down a couple turrets, or have already made it into the base. Even if so, grab the Vampiric Scepter. If you have 1650 or more gold, grab the B.F. Sword instead, then the Vampiric Scepter. After, get their combination, The Bloodthirster. This will greatly increase your Life Stealing rate, because of increased damage, pus the Base Life Steal it adds.
Too Fast? Not Fast Enough! - The Lightbringer
If you haven't won the game already, then might as well aggravate your enemies by increasing your Attack Speed even more. Get the Recurve Bow, or its completed form, The Lightbringer.
In The End
All of these Items work wonders together and alone, but most importantly makes Udyr incredibly powerful and deal severe damage while healing himself and laughing about it.
Since Udyr's Stance's have different Active's, it isn't easily done to benefit from all of them.
In the beginning, grab Phoenix Stance, and then the Bear Stance. Phoenix Stance hurts enemies around you with Magic Damage at a continuous rate, and every third hit deals bonus Magic Damage to nearby enemies.
The Bear Stance is great for stunning enemies, and the temporary speed boost you receive for entering it. Use Bear to stun an enemy in front of your turret, or to hold them in position long enough for an ally or two to help out.
Turtle is only useful in the beginning when your shielding yourself from enemies, or farming minions. In mid to late game, however, you can use the Turtle Shield as a defense, to (almost) completely nullify one blast from a turret.
Tiger Stance, after getting Life Steal and Attack Speed, will quickly replace your Turtle Stance, as you hit faster in Turtle Stance, and 'absorb' Enemy Health based on a percentage of your Life Steal and Attack Damage. My favorite combo would be Bear (to stun the enemy) and then Tiger to increase your attack speed to deal heavy damage in a short time.
Also, if you have plenty of mana and minions, spam the Tiger Stance to get the increased Attack Speed while attacking Turrets. The turret will go down very fast. Just watch for Ganks, stuns, and Enemies who might give chase. You may be flanked and Ganked, and Udyr is not Gank-Proof when alone, no matter how you build him.
What Not To Over Build
Ability Power(AP) - Udyr generally doesn't need Ability Power unless your going to stick with Phoenix the entire game. Even so, enemies only have to add one or two Magic Resist Item's to make your attacks tickle.
Armor - If you build only Armor, you may last longer, but so will your enemies. You don't want them to last very long, because they may have backup closer then you think.
Armor Penetration - This may be good for a going against a noob tank, but chances are, the noob's allies will tell him how to build, and even if they don't, your build most likely won't work if they stack Magic Resistance, or AD or even AP.
Attack Damage(AD) - Udyr's attacks mainly focus on AD for Damage Output, but even so, there's no point in hitting like a truck if you're a feather going against a truck. You may put a dent in their HP, but without some sort armor, you're going down and fast.
Attack Speed(AS) - From what I've heard the maximum Attack Speed is 2 and a half (2.5). I don't know, because every time I tried to find out, my attacks didn't hit high enough, and I had no armor. Lost the match with 1 kill (Kill Steal) and 16 Deaths. Luckily I was against bots in Custom Match, so I wasn't called a noob for looking over different items.
Health - You have tons of Health, but no armor, nor attack. You're just a bigger punching bag with this build.
Health Regen - No matter how much Health you regain a second, if you get hit hard enough, you won't be able to heal fast enough, especially without somekind of protection, or offensive needs. And in Ganks, you'll be done in a second, if you don't run away.
Life Steal - The same reason as AS. If you can't do squat for damage, what's the point?
Magic Resistance - Same with Armor, all they need to do is attack you with AD instead of AP, and you won't do much Damage at all.
Mana - Udyr's Stance's may be costly when using them in a chain, but it won't matter if you aren't doing damage, or softening your enemies attacks.
Mana Regeneration - Woohoo so you got Mana Regen. Udyr doesn't start out with much Mana. Grabbing tons of Mana Regen items will be engraving 'NOOB' on your forehead as you get killed so quick you won't even remembering spawning from your last death.
Speed - Sure, you're quick on your feet, but Boot's don't stack, and you won't be getting much gold while waiting to respawn. Less of a chance of surviving due to no defense or offense.
So basically, don't stack items in just one category, but also don't grab just one item of each and call it good. Udyr has four Stances. He's a man of variety, and so his items should be as well.
Udyr may be looked upon as strong or weak. Slow or fast. But what matters is how you play him. A Champion is only as good as his Summoner. Which is why I'll leave you with this tip.
DON'T FIGHT TEEMO ONE ON ONE! GANK THE STUPID RAT AND MAKE HIM PAY FOR BEING OP TOWARDS THE OP!