The best soil test in the country costs $25 from Logan Labs. Get one of those, post the results here, and people here will tell you what you need (there are different kinds of lime), how much, how often to apply, when to apply, and where to get it. If you use the wrong lime for your soil, it can cause the soil to bind and appear to be like clay.

According to U of Nebraska guidelines a pH range of 5.5-8.0 does not require adjustment. The sweet spot for turfgrass is usually considered to be 6.0-6.5. If you are aerating, you have the opportunity to add lime while the holes are open, getting the lime into the soil profile (lime is very slow to move through the soil). If you are not aerating, maybe consider doing lime in November or December. Surface-applied lime can create high pH at the surface for a while and the winter months when nothing is growing are a good time for lime to gradually dissolve and work in.