Most gym floor mats are made from heavy-duty rubber and thankfully require minimal real cleaning to upkeep. They’re not exactly Persian rugs. That said, make sure you vacuum up chalk fairly often and use a cleaning solution (Simple Green or the like is great) and mop every month — that should be enough unless you’re dripping tons of sweat all over.
During the hotter, summer months, you may want to use the cleaning solution every two weeks instead of monthly. For other specialty gym flooring, refer to the cleaning suggestions provided by the company.