Cloning works by taking DNA from a grown animal and inserting it into an egg cell without any DNA in from another animal. The egg cell is then zapped with electricity to make it grow to a point where it can be inserted into the womb of an animal – this needs to be a the same or very similar animal to the first in order to ensure that the cloned animal will grow properly and can be born.
If we applied this to cloning dinosaurs we would need all the DNA of the dinosaur in order to make sure that we could create it correctly. As dinosaurs laid eggs, the cells wouldn’t be put into an animal but we’d have to find some way of inserting the cloned cells into an egg of a close relative to the dinosaurs that is still alive today, such as a crocodile, or find a way of artificially creating an egg. If we wanted to clone a woolly mammoth, we could put the cells into an elephant as they’re close relatives.