1 hour on separate tickets is risky.
If it is international to international and no checked luggage then just clear transfer security (do not go through immigration & customs).
If it is international to international and luggage is not through checked then immigration (smart gate is quick but doesn't always work), then collect bags (may have to wait for these to arrive) and go through customs (generally short wait but can be longish at busy times). Then to check in for onward flight, but by the time get there it will likely be closed for check in.
If it is international to domestic and no checked bags then as above but also change terminals (10 minute walk away or transfer bus), check in and clear security.
If it is international to domestic and checked bags then no chance of making domestic check in deadline (redeposit bags).