How do we help our child deal with the death of our beloved pet dog?
Your child lost more than a dog, he lost a companion. And to a young kid, things don't get much worse than that. However, this does present a teachable moment for you.
First of all, the loss of a pet is important developmentally as children become practiced and prepared for more significant and inevitable losses later in life: a grandparent, a relative or even a peer. So feeling down and sad for a while is OK.
It's a great time to remind them that relationships are important in life, and because we never know how long they will last, it's important to treat each one as special and valuable. That applies to pets, people and possessions.
And finally, I'd be up-front and honest about the dog "going to a better place." I think it's OK to say something like, "Some people think that all dogs go to heaven, and we'll be able to see him again someday while others don't think so. To be honest, I'm really not sure which is true, but his memory still lives on within us, so we'll always have that to remember." Then give it a few months ... and go get another puppy!