OK, this is my 1st build on here so here we go.
Ive mained udyr for quit a while and I know there's a million ways to jungle him but this is what I find the best. Yes, it is tiger stance simply because Tiger does the same damage as Phoenix just faster so I like it better when ganking early but you can go Phoenix instead.

So this is a little random but trust me it works.
Start with Phoenix Stance, Regrowth Pendant. Go to wolves and use Phoenix twice and move to Wraiths. Smite big wraith, get Turtle Stance, and finish wraiths with 1 Phoenix and 1 Turtle. Next, go to Double Golem. Finish them by using 2 Phoenix and 1 Turtle and you'll hit level 3. (This is where you decide to go Tiger or Phoenix, I like Tiger Better.) Put your skill point into Tiger/Phoenix and recall, you wait 10sec and you'll have just enough to get a Philosopher Stone. Head to Blue Golem and kill it with smite. (easy from here) Then go to wolves, Wraiths, Double Golem, Red Lizard. You should hit level 5 and get a point into Bear Stance (You should have around 3/4 health and the time will be close to 5:25). You can now gank lanes with your double buff.
The Less Detailed Path Is...
(Wolves -> Wraiths -> Golems -> Recall -> Blue -> Wolves -> Wraiths -> Golems -> Red)

Pros -
[*] Early Philosopher Stone (Gold Per 5)
[*] Hard to be counter jungled (Non expected jungle path)
[*] No wasted money on Health Potions
[*] Umm... Its Udyr?

Cons -
[*] Strict jungle to get its potential
[*] Cant counter jungle
[*] Slower than normal level 5 jungles by about 10-20sec

Reds -
[*] Attack Speed (Just the best imo you could go Armor Pen if you'd like)
Yellows -
[*] Armor - Self explanatory (I haven't tried dodge so ill try it later)
Blues -
[*] Attack Speed (To clear jungle faster)
[*] Magic Resist (If you want some extra viability in team fights)
Quints -
[*] Armor Pen (Gives 10 Armor Pen, Jungle creeps have between 9-12)

As I said above starting with Regrowth Pendant allows for an early Philosopher Stone.
[*] Philosopher Stone / Eleias's Miracle - Gold per 5, at the end builds into Eleisa's Miracle for the Tenacity which is needed badly on udyr.
[*] Wriggle's Lantern - Damage, Lifesteal, Armor, and a Free Ward whats not to like. I mainly get it for easy dragons and faster barons. (You could get something in its place if you go Phoenix for its not as effective. Possibly a Hexdrinker?)
[*] Ninja Tabi - I get this over Mercury Treads because I get Eleisa's Miricle for the Tenacity and the bonus dodge chance syncs well with udyr's passive
[*] Wit's End - Great for early game bursting and the Magic Resist is good too.
[*] Randuin's Omen - I use this slot for armor heavy items, I prefer Randuin's Omen for the massive armor and the active slow.
Now here is where your choice/enemy team comp comes into play.
[*] Force of Nature / Banshee's Veil / Sunfire Cape - This is the late game Magic Resist item. I like Force of Nature for the movespeed and heavy Magic Resist, but for CC heavy teams Banshee's Veil is perfectly acceptable. AND, if their AP champs aren't effecting you and you would like some more armor/health Sunfire Cape would be your best bet.

Smite is necessary for the jungle (Derp) so the other slot is up to you.
I like to take ghost for the easier ganks. (Ghost -> Bear -> Tiger is usually a kill)
Flash could be taken if u prefer it, but i just think ghost is more reliable.
I wouldn't take any other than Ghost / Flash, I suppose Cleanse could be taken but I'd rather have an initiative / escape summoner but that's just me.

Not too much to say here besides I go 21-0-9 for the burst from offense while taking upgraded smite and ghost.
The rest is self explanatory and I haven't tried this jungle without it so TAKE IT.

Not gona lie I don't know if I posted the correct sequence but here it goes in words.
Start with Phoenix, Turtle, then your choice either prioritize Phoenix or Tiger.
Sorry if it sounds complicated i cant really tell how it is.
Phoenix -> Turtle -> Tiger/Phoenix -> Tiger/Turtle -> Bear
And so on maxing out Tiger or Phoenix (your choice) then Turtle then Bear and leaving whatever you didn't max out 1st (Tiger or Phoenix) last and only getting 3 levels.
[*] Tiger - I max this out 1st because of the insane burst damage.
[*] Turtle - I max this out 2nd for the shield and health/mana regen.
[*] Bear - You HAVE to get one point into this at level 5 for ganks and just to be udyr.
[*] Phoenix - You could max this out instead of Tiger your choice. (It has the same damage as tiger just not upfront, more in the AoE and is awesome for farming creeps.)

All in all this is my way of jungling udyr the point is that its MY way so you don't have to use it but try it and if you don't like it then I'm open to constructive criticism.
And again this is my 1st so please no hate.