He wouldn't replace Knox in Washington's administration; Knox was Washington's artillery officer, they had worked together throughout the war and genuinely liked each other. The early republic being both conservative and not really stable, they're going to give Adams a go rather than switch to Greene, and Adams won't change the Cabinet.
There are others who know Greene better than I around here, but I assume he was a Federalist; perhaps Adams would want him for VP so he runs in '96 but the slot goes to Jefferson; but in 1800 everyone's sick of Adams but now Federalism has a viable replacement for him?
I'm not sure if Greene can have the momentum (and he might be too old by 1800), but that's where I'd expect him to wind up: as Jefferson's foil. He might also be able to shift some Southern electoral votes toward Federalism.
I suppose you could have Knox die, of course. Greene would be a good replacement for Knox...