There are at least 5 ways to de-extinct a species. Here they are, arranged from least likely to result in de-extinct non-avian dinosaurs to most likely.
1. Iterative evolution
Also known as Elvis taxa, this is a form of very precis convergent evolution. A species goes extinct, possibly because the environment where they lived changed and their niche disappears. Later, that niche became available again, and a new species evolves to fill that niche. The new species looks nearly identical to the original species.
This happened with the dinosaur (flightless bird), the Aldabra rail.

Presumably, the Aldabra rail evolved from relatives in Madagascar. They lived there until about 136,000 years ago.
Then sea levels rose, and their island went underwater. The Aldabra rail went extinct.
Eventually, the sea levels dropped. And it appears a new population of rail arrived from Madagascar. They evolved flightlessness the same way as the original Aldabra rail, and they look very similar to the Aldabra rail.
2. Breeding back
The gist is animals are selectively bred to look or act like an extinct species. An example of this is modern cattle, which were domesticated from aurochs.
Since we don’t have anything with dinosaur traits we’d like to re-create, this isn’t really feasible for dinosaurs.
3. Cloning
This is the only de-extinction option that results in a genetically identical species. It requires an intact nucleus of the extinct species. The process involves removing the nucleus, sticking it into an ovum of a related species, and having the related species do the de-extinction work.
However, no bird has ever been cloned because of how the egg moves and develops a hard shell in the oviduct.
Also, this is not going to happen with a non-avian dinosaur, since we don’t have an intact nucleus.
4. Genome editing (specifically by using CRISPR/Cas9 system)
For de-extinction, genes can be incorporated that make an animal unique into a living relative’s DNA. This allows the use of degraded DNA, since you only need to pull out some of the genes.
But, it’s very unlikely an non-avian dinosaur DNA has survived to the point where we could get enough fragments to make this work.
5. Activating and inactivating genes
This is similar to breeding back, but involves directly modifying which genes are expressed. Animals often have unexpressed genes of their ancestors, or they are expressed in different amounts.
Paleontologist Jack Horner has talked about using this method to create a “Chickenosaurus.”
This de-extinction technique would likely work for non-avian dinosaurs, thought it would probably only work for theropods.
Check out our podcast, episode 332, to hear even more about techniques to de-extinct a species.