The good news is that there were a lot of historians and chroniclers active during Belisarius’ lifetime, and most of them were aware that they were living through significant times. Procopius stopped writing in 552 and was continued by Agathias Scholasticus till the year 588. He was in turn followed by a man named Menander Protector who decided to write history after he had blown his life savings having a good time. This was evidently a smart career choice as Menander was soon joined by a crowd of writers. If you fancied Justinian’s reign you could pick up bestsellers from Evagrius, John of Epiphaneia, John Malalas, and Marcellinus Comes.
The bad news, however, is that most of these are only partially translated today- or worse exist only in fragments. Unless you know someone at Dumbarton Oaks, you’re probably not going to be able to read them anytime soon. The only commercially available, non-Procopius source is Jordanes. He was a 6th century Roman bureaucrat who had a book collection and the rare gift (as he put it) of being concise. A friend who wanted to read a history of the Goths asked to borrow a copy of Cassiodorus’ (now lost) work, but when he saw its size (12 volumes) he asked Jordanes to sum it up for him. The result is the Getica, which concludes with Belisarius’ brilliant defeat of the Goths in 540.