Once you have the dog key (see the guide here)
work your way towards the end of the game, right up to the point where you watch the video tape in room 312 at the Lakeside Hotel. After watching the video, use the dog key on the door directly next to room 312 (Observation room) to see the dog ending.
This is also the best ending to go for if you're trying to beat the game in under 3 hours, or if you're going for any of the harder achievements, as the game will end sooner (as soon as you use the dog key)