As is indicated in the film, native Kryptonians are born genetically pre-ordained to certain roles within their society based on the codex, and for Zod that was to be a military commander. Zod not only excels in this role, he relishes it, it is every fiber of his being and existence. His single purpose in life is to protect Kryptonian society, and bring those who break its laws, fail to do what is best for it, or who seek to destroy it to justice as he sees fit, and in his eyes Jor-El is a heretic, a thief of the highest order, and his son the illicit and illegal offspring based on Kryptonian law, in addition to stealing the item he needed most from his coup: the codex.
His motivation isn't just spite for Jor-El and a desire to uphold Kryptonian law, there's also a sort of genocidal aspect to Zod's endeavors. Zod wants the codex so he can birth new Kryptonians who will follow his orders and uphold the version society he feels is best to continue the Kryptonian race, but since Kal-El isn't the product of the codex, Zod views Kal-El as genetically and therefore racially inferior. Given he's spent decades living among humans who have largely treated him as one of their own until certain recent events, coupled with them having virtually no way to stop him given his advanced technology, he likely views the terraforming and therefore "destruction" of earth as the ultimate solution to both his vision and his vengeance. Considering earth is also within what's referred to as a Habitable Zone and occupied predominantly by a species they have no feelings for and can easily destroy, why not just take the planet?
