Dinosaurs were around for (very roughly) 3.68% of the Earth’s current time span.
Our best estimates for the origins of dinosaur evolution are approx. 233 Ma (millions of years ago) during the Ladinian in the middle Triassic Period. The estimation of the time of their extinction (known as the Cretaceous-Paleogene extinction event) was about 66 Ma which brought about the end of the Cretaceous Period.

This image illustrates the basic geologic timeline with different evolutionary properties of various organisms.
So as for the percentage: if the Earth is 4.54 billion years old and dinosaurs existed for 167 million of that (233M – 66M = 167M) then we get:
##167000000 -: 4540000000 = 0.03678##
Take the number we get from that and multiply by 100 to get a percentage.
##0.03678 * 100 = 3.678%##
Round off to 2 decimal places to get 3.68%. So dinosaurs were around for 3.68% of the time since Earth’s origin!