How do I replace my electrical receptacle?
The fact that four neutral (white) wires are connected is not a problem, but the method of connection is. Putting two wires under a terminal screw is not permitted (even if often done) and can lead to a bad connection.

The proper way to connect multiple wires in a box is to connect two or more with wire nuts or the newer push-in connectors (follow wire nut/connector limits on number of wires). While two neutral wires can be attached to the same side of an outlet and this provides connectivity, it is generally considered a poor technique, since a failure in the outlet may disable a downstream device.

In a box with two cables, the preferred way to connect the neutral to the outlet is with a pigtail. Join a short white wire with the two other whites and connect with a wire nut. An additional neutral can be added if there are three cables or a second outlet is present. For more wires, you can join wire bundles with pigtails so long as you do not exceed the required box volume limits.

Hot (black or red) wires can be similarly bundled with wirenuts and pigtails, but this occurs slightly less often because black wires are often separately routed to and from switches. 

Connect your neutrals using wire nuts and a pigtail to the neutral side of the GFCI. Connect the hot directly to the outlet.