I've always viewed a clone as being comparable to an identical twin. An identical twin strangely conceived and born at a later date through artificial means, but an identical twin nonetheless.
To that end, I don't see why a person's clone would be deemed to be less "real" than a person's identical twin or why a person's clone would have the same personality, memories, etc as the person he or she was cloned from rather than developing a personality, memories, etc of his or her own.
Moreover, although I frankly don't believe in souls myself: I don't see why a person's clone would be any less worthy of a soul of his or her own than a person's identical twin would be either, if souls hypothetically existed.
If you were to clone your dead child, you would have a new child that's genetically identical to the first and would develop his or her own personality and memories (and would hypothetically have his or her own soul). I don't see anything wrong with this per say, especially if the parents cannot conceive another child for some reason, but I would not treat this clone as my deceased child. I treat this clone as a new child with a new identity altogether.
Cloning would not be a route to immortality unless you were to create a mindless shell body that you could then transfer your current mind into it and occupy. However, unless such a shell were mindless from the start and somehow prevented from developing their own mind prior to transfer: you are essentially overwriting your twin's personality in an abstract form of murder. Putting even that aside, cloning could prolong life if we were to grow cloned replacement parts--though harvesting parts from clones is also another, more morally questionable given the above, alternative.