By looking at ancient proteins, amber and bird genomes, it is possible to deduce what the genome of some dinosaur species looked like. However, chances are that it would not be possible to find enough paleo-DNA to clone the whole animal (Source 1).
However, this does not mean that cloning dinosaurs is impossible. Scientists search for paleo-DNA because they do not fully understand how DNA creates an organism. However, one day scientists will understand it. Because of this, looking for ancient DNA and proteins will no longer be needed. It would simply be possible to create it with a computer (Sources 1 and 2).
Lee M. Silver investigates this in his book Challenging Nature. In the pages 312-313, Silver argues that by using a digital chicken genome, it would be possible to deduce and create the genome of a non-avian dinosaur (Source 2).
"...evolutionary developmental geneticists believe that comparisons of DNA in birds and reptiles -on either side of the evolutionary line leading to dinosaurs- could provide some, although not complete, insight into the extinct genome of dinosaurs. Genetic, developmental, and evolutionary understanding and data could be combined with molecular understanding to allow rational design of a dinosaur on a computer. Starting with a virtual chicken genome, virtual genetic changes could be implemented to increase overall adult size, eliminate feathers and restore scales, turn wings back into elongated front and back legs, and mold the virtual animal into a reasonable facsimile of a particular dinosaur, such as a giant long-necked apatosaurus, a triceratops, or even a Tyrannosaurus rex. The designed electronic genome would be converted into organic DNA with nano-DNA writing machines that automatically link up thousands of smaller DNA fragments into whole chromosomes. The genome would then be inserted into a chicken egg devoid of its own DNA -in an advanced version of the process that created Dolly- and presto, dinosaur facsimiles."
Using this method, it might also be possible to recreate other extinct animals, such as marine reptiles and pterosaurs.
2. Challenging Nature: The Clash of Science and Spirituality at the New Frontiers of Life.