A question. Where are you located? Soil structure will improve itself naturally under certain conditions. Freeze/thaw cycles cause the ground to "heave", breaking compaction. That's one of the keys to successful no-till farming. It CAN remedy your situation in time. Not so if you live in a southern climate where freezing doesn't take place often enough or hard enough to do the job.
Call your county extention agent. They can help you estimate degree and depth of compaction. They can ALSO give you more localized advice. Local conditions effect "what works best".
Grass, like ANY plant, doesn't like hard compacted soil. At the same time, grass doesn't need, nor does it like to be seeded into loose, fluffy soil. FIRM soil is best.