How do I replace 4 GFCI receptacles on the same circuit?
This configuration is a bit odd.  The reason why a GFCI outlet has a line and load side is that a single GFCI outlet can offer GFCI protection to any downstream outlets (load).  

If I were in your situation, I would install a new GFCI outlet in the first location, and then new standard outlets (your region might require tamper-resistant outlets) in all the downstream locations.  This should make troubleshooting simpler, plus you only have 1 GFCI to deal with.

Alternatively, if you do wish to install a GFCI in each location, they should not be wired to the previous outlets load side, rather they should be wired in parallel like standard outlets (either a pig-tail or using the second screw on the same side of the outlet).

If after correcting the installation it is still tripping, then you have a current leak somewhere.  A GFCI compares the current on hot and the current on neutral, and if the two are different by a certain threshold, the breaker trips.  It could be a device plugged in somewhere else on the circuit, a nicked wire, etc.  Start by unplugging everything on the breaker.  Plug something into the GFCI outlet - it shouldn't trip, if it does, it's a wiring problem or the outlet itself is bad (if you just replaced it, this shouldn't be the case).  One-by-one, start plugging in other devices upstream and downstream of the outlet. When/if it trips, you have found your problematic device or outlet.