How to find what tables are stored in what data files
You can find directly which rowset (partition of an index/heap) belongs to which filegroup: look at data_space_id in sys.allocation_units. But is not going to give you details at the individual file-within-filegroup level. It is possible to find exactly how each rowset page lays out in each file by chasing the IAM chains manually, or by using undocumented features like DBCC IND or %%physloc%%. Not trivial, but possible.
But I would encourage you to go down a different path. Specially in the light of your comment about correlating space used in the file with IO, since the largest file by size does not necessarily imply most IO in the file (think data at rest). Instead, use sys.dm_db_index_usage_stats and, especially, sys.dm_db_index_operational_stats instead. You will get the load on each rowset, and then you can correlate this with the actual physical IO.