how to play dvd on my computer on apple tv

Welcome to the Apple Community Jorge.

Firstly you will need your Mac to support Mirroring...

AirPlay Mirroring requires a second-generation Apple TV or later, OS X 10.8 or better and is supported on the following Mac models: iMac (Mid 2011 or newer), Mac mini (Mid 2011 or newer), MacBook Air (Mid 2011 or newer), and MacBook Pro (Early 2011 or newer). You can see which Mac you have and which operating system you are using by selecting 'About this Mac' from the Apple menu in the top left corner of your Mac and selecting the 'More Info' button.

AirPlay Mirroring from a Mac also requires a fairly robust network.

Then you will need to use a 3rd party DVD player such as VLC since the Apple DVD Player software doesn't permit AirPlay Mirroring.

I am having the same problem. I have a Macbook Pro Late 2011 OS X 10.9.2 and an Apple TV Model MC572LL/A (2nd gen) Software 6.1 (6698.99.16). When I use AirPlay with the DVD player, the movie appears for a moment on the TV only while resizing the screen, but then goes to a blank grey screen. Audio can still be heard. Please help.

As stated above the native DVD player blocks airplay mirroring. You will need a 3rd party program.

Just download a third-party player like VLC. Its working this very moment I'm typing this out! 🙂

You are responding to a three year old thread where that was already stated.