"One white" limits possibilities. Provided that you don't have some monkey business downstream it would seem that your circuit is a "multiwire branch circuit" that at the location you are working the halves of the receptacles are fed by separate 120v circuits, and the black and red that have no voltage sensed on them feed a downstream 240v outlet (as allowed by NEC 210.4(C) exception 2). Both of these should be fed by adjacent handle tied breakers or a two-pole breaker.
Your problem is the standard style receptacle has break-off tabs that allow the top and bottom to be fed separately, GFCI receptacles do not have this feature. If GFCI protection is required you have two options. (1)Protect all the involved outlets by connecting the existing style receptacle wired as it originally was, with a broken hot tab, and feed it with a two pole GFCI breaker which would maintain availability of both circuits at this location [that option won't work if somebody has improperly acquired a neutral downstream from another circuit or put some current on the ground], or (2) Pick only use one circuit for use to feed the GFCI receptacle by using a wire connectors to connect the blacks together with a third wire to brass line screw on the receptacle, the two reds together with another wire connector, and the white wire directly to the silver line screw.
The load screws are unusable with either configuration.
A ground wire needs to feed to the receptacle, the feed through to the 240v downstream feed, and to the box if metal.