Use this information at your own risk.
Find some intact dinosaur DNA.
Dinosaur blood sucked up by a mosquito that was then immediately preserved in amber should do it. Such specimens would be rare, but let's just say you find one.
Though amber is a great preserver, it doesn't stop DNA degrading over time. A study published last year in the journal Proceedings of the Royal Society B measured the half-life of DNA in 158 radiocarbon-dated bones and found the average was 521 years.
Which means that every bond in dinosaur DNA would be broken after just 6.8 million years. Given that the last (non-avian) dinosaur died about 65 million years ago, that's bad news for anyone hoping to find any intact T. rex DNA at all.
Another study showed that DNA is unlikely to be preserved in copal (a type of tree resin), meaning it's probably even more unlikely to be preserved in amber, which is millions of years older.
Extract the DNA.
Putting aside the fact that there'd be no DNA left to speak of, it'd be almost impossible to extract it from inside a mosquito without mixing up some of the insect DNA in there too.
Nobody wants a half-stegosaurus, half-mosquito.
Sequence the dinosaur's genome from the DNA.
Before you can think about cloning an organism, you need its complete set of genes, or genome. You're extremely unlikely to get that from whatever DNA you've extracted from the mosquito, so you're going to have to improvise.
If we were to clone a mammoth or neanderthal, we could use an elephant or human's genome to work out where each scrap of DNA goes and start reconstructing the whole genome.
In the film of Jurassic Park, scientists fill in the gaps with frog DNA (the book mentions using the DNA of other animals, too).
But actually the best candidate for filling in dinosaur DNA would be birds, says Brian Switek at Mental Floss. They are the only living things left over from the time dinosaurs roamed Earth. Even so, their use in helping us figure our dinosaur genomes would be "quite limited".
From this patchwork genome you then need to construct some chromosomes. Sadly, we don't know how to do for dinosaurs.