Which one do you prefer? I myself think that an AD Udyr is better but leave a comment on what you think or any kind of changes you think are needed. AD udyr has many pro's and few cons.
Pros:
1.large amount of health
2.good armor
3.good damage
4.decent attack speed(especially during tiger stance)
5.large health regen(from warmogs armor and life steal)

Cons:
1.4th move is useless
2.not that fast
3.not a huge dps

AP udyr is just one i haven't used much. I always use AD so i have little experience with AP udyr. Some pros and cons i know is...

Pros:
1.massive ability power
2.decent armor
3.decent magic resist

Cons:
1.low hp
2.low mana
3.only effective move is phoenix

AD and AP udyr both have specific items.
AD focus's on health so you can get the most amount of damage from atma's impaler and get a great amount of armor. Banshee's Veil helps take the least amount of damage possible by blocking abilities.
AP focus's on massive ability power and attack speed. With the massive ap it makes phoenix form do some massive damage on active and decent on passive.

Lore:The mystics of Ionia delve into the deepest mysteries of Valoran, pioneering the most inner depths of spirituality. Their adepts are some of the most devout proponents of enlightenment and harmony...but as is always the case, there are those who follow a very different path. Born in a grassy clearing under a red moon, Udyr has always been compelled by the primal drive within him. His will was not untempered, simply undomesticated. As a boy, he toiled on his father's farm, but he felt a much deeper connection with the mustangs that would graze in the distant fields than with the townsfolk who would visit. Oftentimes he was dragged into the house at night with a swat and a scolding when his parents would find him sleeping beneath the stars.

On the day of his sixteenth birthday, Udyr bade his family farewell and struck a path to the east, determined to shed the trappings of ''civilized'' society. Freed of the shackles of cultural expectations, Udyr tapped into a well of inner ferocity he hadn't known before. His feral senses roiled to the surface and he let them overcome him. It was at this point in his life that Udyr truly lost his humanity. Poachers and travelers who entered his territory did so at terrible risk, and rarely managed to leave it.

A monk on a long journey passed through his woods one day, and Udyr decided to scare him, leaping from a nearby brush. The monk casually turned and redirected Udyr to one side. Udyr, enraged, tried time and time again to best him, but the monk was unassailable. When Udyr was exhausted, the monk wordlessly beckoned him to follow, and together they walked in silence to the Hirana Monastery. The monks there took him in, and taught him to harness and control his animalistic fury...most of the time.