I go straight for Wriggles and then upgraded boots with alacrity. This gives you so much speed to run around and farm with. After that either tank items if you want to be a team player or zephyr and trinity if you want to keep farming and split push.
You have play with a screw my team attitude early though. Your farming early is more important than ganking. I've had bot lane rage on me as Eve went up 4/1/4 ganking them, but I know going down there is going to slow down my ability to power spike and carry. 25mins into the game my team wins via surrender as I had 4 levels and 5k on Eve off all that farming while she ganked.
Go to Youtube and watch Trick2g's Godyr 4.5 videos. He pretty much preaches if you want to carry at lower Elo's which is pretty much Gold and below you have to Ignore, Farm, and Pressure. He typically maxes Phoenix and Bear for the lower mana cost and movement speed. This lets you fly around the jungle getting farm. Wriggles and alacrity boots are a must as you want all that movement speed. Speed equals faster farming and more pressure as you can move around the map faster than they can keep up with you.
He likes to go for Sunfire Cape as a 3rd item, but after the first 2 you can really do whatever you want. If you want to start scraping more often with you team then getting tanky is recommended. If you want to keep farming and start split pushing then go with Zephyr like I said above.
At level 11 you want to take a point in Tiger if you have not already. Udyr with Zephyr and Tiger can create a ton of map pressure, especially at lower Elo's were warding is non-existent. Tiger > Phoenix = Goodbye Minions. Turret meet Tiger = Goodbye Turret. Anyone shows to stop you = Run, Run, Run Fast as You Can You'll Never Catch Me I'm in Bear Stance Man.
If you farmed well and they send 1 person into the lane you're pushing you can smack them in the face, chase them down, and kill them. It likely won't matter who it is because you should have a level advantage if you farmed well. If they send the whole army you just run away into your Jungle and take a few camps before going right back to pushing the lane. You do want to be using your free ward from the Flare and Trinket though to have vision on your flanks so you can run away. It's also never a bad idea to grab an extra ward at the shop. The only thing that is going to stop you is hard CC and getting jumped on unexpectedly.
If you go Tanky you want to start living in their Jungle and looking to flank and pick people off with your team. Tanky means you're less about split pushing and running away and more about looking for fights.
Biggest thing to playing Udyr well though is pressure, pressure, pressure. He is not a let me group with my team and run down mid jungler. He is a make them come and try to stop me champion. If you just Bear Stance and run straight at a grouped up a team you will die. They will kite you, CC you, and kill you. If you're taking a tower or pushing a lane though they have to come to you. This makes it much easier to close the gap and get a fight in your favor. You'd also much rather show up mid-fight than be around for the start of it. Once a fight has started and abilities have used and team members have been split up, that is when Udyr wants to show himself and start running people down.
1 Champion putting CC on is not going to stop Udyr, especially if Flash is up. It's when one or two champions hit you with CC while one or two carries get in free shots that kills the Manbear. When you show up mid-fight it's highly unlikely you get that kind of focus put on you unless your entire team is dead at which point you shouldn't have gone in anyway.