Lens for Nikon D 5100 camera?

I have just bought a Nikon D 5100 camera. I'm now looking to buy a telephoto lens for the same. I wanted to know which would be the ideal lens for taking wildlife photographs and photographs of my children.
Do i need to buy Nikor lenses only or do other lenses work OK?
How important is the auto-focus being inbuilt in the lens. As my camera does not have a AF motor on the camera, it needs to be on the lens. So is an AF motor necessary on the lens?

To get autofocus you want a motorized lens. Nikon AF-S, Sigma HSM

For wildlife you want something long, cheapest option is Sigma's 150-500

As an allround tele you can make do with something that reaches 300mm. I like the nikon 70-300 VR because of it's build quality but the 55-300 may be an interesting budget option as well