Udyr is a fairly strong duelist in the early game, winning almost every 1v1 thanks to his Tiger
Stance. Avoid solo fights with him: focus on farming and warding jungle entrances to avoid Invades.
Udyr struggles clearing multi-monster camps such as Raptors or Krugs early. Use this to your advantage and steal them when possible or invade him while his trying to clear them.
Udyr ganks are very straightforward: run to you, stun you and shred your health down with his Tiger Stance
. Keep your river bushes warded and don't hesitate to Flash if he comes too close.
Udyr has no reliable engage tool other than running towards your team. Kite him so he is unable to land his CC onto your carries.
If he ever uses his Flash to engage, remember that he has no other escape tool. Focus him down as quickly as possible.
In team fights Udyr will usually try to 100-0 someone with a Flash and a full rotation of spells. If you can survive his first rotation and get away from him- there is nothing he can do to close the gap. Kite him down while retreating.
At level 3, Udyr gets access to his first full rotation, keep in mind that he needs 6 seconds to use a stance again. Use the windows where he has one Stance down to trade with him as it reduces his chances of surviving.
Once he has his Trinity Force, Udyr becomes very difficult to fight alone. Team up with a buddy to take him down 2v1.
Once Udyr gets a few items behind him, he is able to solo the Baron with ease. Keep this in mind and make sure you always have the Baron warded post 20 minutes.