The Turkish threat was very real, and a mighty problem for the survival of some of the christian states of the Mediterranean, particularly Venice, but the Holy See was not particularly secure either. Hence, the creation of a christian alliance was not unreasonable in any way. Throughout the XVI century, there were Leagues formed against the Turk at different points in time, and the composition was not significantly different than the League from 1571.
For instance, the negotiations for the creation of the Holy League in 1537 had the implication of the Pope, the Doge of Venice, Spain, and Andrea Doria. This is more or less the same lineup as with the Holy League from 1571. The most interesting document on this respect concerning the Holy League of 1538 is this one, from the General Archive of Simancas. In there, you can read, if you speak Spanish all sorts of details concerning the negotiations.
AGS, EST,LEG,1370,214. Letter to Andrea Gritti, Dux of Venice, concerning tbe creation of the Holy League against the turks by Spain and Pope Paul III.
There are more documents that show the complexity of the multilateral negotiations, particularly in the AGS, section Estado, shelfmarks 1370, and 1371. You can find them digitalised through PArEs.
Back to the Holy League of 1571, the fall of Cyprus, and the precedent of the Siege of Malta, made the christian states very much aware of the threat that was the Great Turk. Knowing that there was a precedent, and that all parts were Catholic, things could be smoothed over. Under the Holy League, the command decisions would be taken by consensus, but the Captain General would be John of Austria, bastard brother of Philip II. As things would go in order of importance, so would be the expenses of the war, and the war effort.
Spain would put half of the money and military effectives, both men and ships.
The Holy See would put 1/6, and Venice would foot 2/6 of the cost in men, money, and ships. Genoa would contribute to the Spanish fleet.
As for how was the alliance maintained, it wasn't. It collapsed about a year after the great victory of Lepanto.
I have tried to be brief and explain as many things as possible.

Sources:
AGS, Consejo de Estado, ledgers 1370, 1371, 1402.
Braudel, Fernand, The Mediterranean and the mediterranean world in the times of Philip II
Abulafia, David, The Great Sea, London, Penguin Books, 2014. Pages 411-418.
Doria, Giovanni Andrea, Vita del principe Giovanni Andrea Doria, scritta da lui medesimo, Genova, Compagnia dei Librai, 1997.