As J-Force said, there's probably no such thing as an "unbiased" history of the First Crusade (your preferences will depend on which side you're already biased towards). But here are some other possible choices:

Thomas Asbridge, The First Crusade: A New History (Oxford University Press, 2004)

John France, Victory in the East (Cambridge University Press, 1997) (for a military history perspective)

Peter Frankopan, The First Crusade: The Call from the East (Harvard University Press, 2016)

Jonathan Riley-Smith, The First Crusade and the Idea of Crusading (University of Pennsylvania Press, 1986) (as CSLeoK mentions, Riley-Smith is a bit old and out of date now)

Jay Rubinstein, Armies of Heaven: The First Crusade and the Quest for Apocalypse (Basic Books, 2011)

Robert Chazan, In the Year 1096: The Jews and the First Crusade (Jewish Publication Society, 1996) (for the Jewish perspective)

There are also lots of collections of primary sources, such as Edward Peters, The First Crusade: The Chronicle of Fulcher of Chartres and Other Source Materials (University of Pennsylvania Press, 1998)

To be honest, there aren't many good books about the First Crusade. I don't think any of the currently available books dedicated to the topic of the First Crusade meet the criteria of 'reasonably in depth and unbiased as possible'. Many meet the first criteria, I do not think that any meet the second. You've got the great crusader historians like Runicman and Riley-Smith, but they are hopelessly biased and are now decades out of date.

These days one of the best books on the Crusades in general is Christopher Tyerman's God's War. It covers all the crusades, but runs to nearly 1000 pages so there is plenty of detail about the First Crusade. It isn't very well written though - his intense knowledge of detail and extremely wide vocabulary can bewilder readers. But importantly, he doesn't take sides. For some reason, writers on the First Crusade have a tendency to be really pro-crusader or anti-crusader, and, in the 21st century, pro-Islam or anti-Islam. Tyerman has far more nuance than that and I think he is perhaps the only author to have written a detailed account of the First Crusade who manages to do it without serious bias. It's a difficult read, but you'll know far more from it about the Crusades than any other single book.