There are really only three histories that cover the entire campaign. They complement one another, because each takes a different approach.
Hess, Earl J.. In the Trenches at Petersburg: Field Fortifications and Confederate Defeat. Chapel Hill, N.C.: University of North Carolina Press, 2009. This book takes the most original point of view and focuses on the fortifications. The author would have done well to study siege warfare during other periods, but just the writing about the fortifications is worth the price of the book. One of the book's insights is that the field fortifications delayed Confederate defeat in that Federal troops who should have been training to fight spent their time digging instead.
Horn, John. The Petersburg Campaign: June 1864-April 1865. Conshohocken, Pa.: Combined Books, Inc., 1993. An interpretative, analytical, traditional approach--except for excerpts from the diaries of Pvt. George S. Bernard of the 12th Virginia Infantry, the Petersburg Regiment, at the heart of much of the fighting Southside (south of the Appomattox River). Long on tables, short on maps.
Trudeau, Noah Andre. The Last Citadel: Petersburg, Virginia, June 1864- April, 1865. El Dorado Hills, Ca.: Savas Beatie, 2014. An eyewitness history of the siege.
There are two histories that focus on the fighting Southside, a very substantial part of the siege. Because one of these histories focuses mainly on the Federal side, and the other on the Confederate side, these volumes complement each other.
Bearss, Edwin C., with Suderow, Bryce A. The Petersburg Campaign. 2 Vols. El Dorado Hills, Ca.: Savas Beatie, 2012. These volumes build on Mr. Bearss' studies of the main actions Southside. They focus mostly on the Official Records and the Federal side of things.
The other history has quite a history of its own.
Bernard, George S., ed. War Talks Of Confederate Veterans: Addresses delivered before A.P. Hill Camp of Confederate Veterans, of Petersburg, Va., with ADDENDA giving Statements of Participants, Eye-Witnesses and others, in respect to Campaigns, Battles, Prison Life and other War Experiences. Petersburg: Fenn & Owen, 1892. Bernard and his comrades would write articles, publish them, seek comments from their fellow soldiers, and publish the results. This volume is critical to the understanding of the battle of the Crater, among other fights--not all of them around Petersburg.
Newsome, Hampton, Horn, John and Selby, John. Civil War Talks: Further Reminiscences of George S. Bernard and His Fellow Veterans. Charlottesville, Va.: University Press of Virginia, 2012. Bernard originally compiled and edited this book. He was ready to publish it in 1896, essentially as volume 2 of War Talks Of Confederate Veterans. Something happened and the manuscript disappeared. It showed up at a flea market in 2004, where it was purchased for fifty dollars. Later it was sold to the History Museum of Western Virginia for fifteen thousand. This volume covers the battles of Globe Tavern (August 1864) and Burgess Mill (October 1864), among others--again, not all of them around Petersburg.