For general Crusades history I recommend:

Thomas Asbridge The Crusades, Jonathan Riley-Smith The Crusades, and Christopher Tyerman's God's War.

Both Asbridge and Riley-Smith have written histories of the First Crusade specifically. I generally prefer Asbridge (he's a little more recent, and I prefer his writing style), but both are great historians.

For another perspective, Peter Frankopan's The First Crusade: The Call From the East is an account based on the Byzantine perspective.

There aren't any good Muslim accounts of the First Crusade, mostly because it is barely mentioned in contemporary chronicles. However, Paul Cobb's The Race for Paradise is a great Islamic-focused history of the crusades generally, and includes some useful details on the First Crusade.

You might also want to look into the Penguin Classics Chronicles of the First Crusade. It's a narrative of the First Crusade cobbled together from several of the major Latin chronicles.