For one, the monster has to have a plausible origin story (since what makes kaiju interesting is that they rampage on Earth). This is different than a zombie story, which can just happen "as is" with no explanation as to its origin. People that are into kaiju films are the same kind of people that like their fantasy with well-detailed magic systems where an author goes to great lengths to provide explanations of cause and effect. I think if you don't explain the kaiju well, then the whole story kind of falls apart like a bad souffle.
I admire how Pacific Rim handled this: creating an alternate dimension filled with hostile beings that used genetically-crafted giant monsters as their weapons of mass destruction. In a word? Brilliant. Godzilla is equally so because the monsters ended up being primordial beasts that fed on the radiation of a young Earth. As their food diminished over time, these kaiju were forced to dig deeper into the earth to search for it (and of course most of them died off).
I also think that there are certain rules that all would-be kaiju creators should obey. 1) They must have some kind of unique super power that is used sparingly but is pretty kick ass when employed. 2) They must be really difficult to kill. 3) They must be terrifying. 4) They have to destroy at least one human city.
With this in mind, I think I need to figure out if my kaiju is going to come from a different dimension, from outer space, or from Earth.
Anyway, enough of my musings. If you can't tell, I loved Godzilla (psst...he does have atomic breath). You would be wise to go and see it if you are a fan of huge monsters destroying cities. Have a great weekend.