First you'd have to explain why you have the British and French on seperate sides, how you get a US/German alliance and how Russia and the US, who were very close until the Communists took over, became estranged.
What if the Franco-Prussian War is less conclusive, leaving Prussia as sovereign and not as dominant?
Additionally, Russia has an earlier revolution--maybe the new government refuses to recognize a tsarist sale of Alaska to the US (probably later than 1867). A less overpowering Germany means that Franco-British disputes outweigh their commonalities when the war comes.
So France and Britain are at odds, France and Prussia are at odds, and Russia and the US are at odds. A Franco-Russian alliance is not inconceivable; the hard part is getting Austria onto their side as well. Austria's Balkan ambitions include Russia's little Slavic brothers. Maybe Italy is a close British ally, as well as Prussia; the Ottomans likely have British backing against the Russians. Austria-Hungary might decide it would get along better with France and Russia than with Italy.
With this, the alliances are ready, except for Spain and Sweden; their purposes as members of their alliances might be to threaten Gibraltar and Finland, respectively.