Anastasius to Justin
When Zeno died in 491 AD, his widow Ariadne chose one of Zeno's advisors, Anastasius, to become the new emperor (and she married him too). Zeno had left the Empire in good shape, with plenty of money in the treasury. Anastasius was an older man when he became emperor, in his early sixties, and he was very careful with money, so that the treasury did very well under Anastasius as well, though he got a reputation for being a skin-flint (a miser).
Anastasius fought a long war with the Sassanians which ended after three years with both sides so tired out that they agreed to just give back whatever they had conquered and have everything the way it was before the war began.
Anastasius lived to bevery old - nearly ninety! - and had a long reign and a long marriage to Ariadne. Ariadne died in 515 AD, and Anastasius followed her three years later in 518 AD, without any sons. Justin, who was the captain of Anastasius' bodyguards, made himself emperor, and announced that he had been Anastasius' choice, though that's probably not true.
Justin, like Anastasius, was also an old man when he got into power. He was about 70 years old when Anastasius died. Because he had always been a soldier, people said, he didn't know how to read or write. Justin didn't know much about running an empire, so he depended on his advisers, who luckily did a good job. One of Justin's main advisers was his nephew Justinian, and when Justin began to get sick, in April of 527, he named Justinian as his successor. Justin died in August of the same year, and Justinian became emperor.
Oxford Children's History of the World, by Neil Grant (2000). A general history of the world for kids. Good place to start.
Early Medieval Europe, 300-1000, by Roger Collins (1991, 2nd edition 1999). Accurate and well-written.
History of Byzantium: 306-1453, by Timothy Gregory (2005). Also for adults, but more up to date.
Constantinople: The Forgotten Empire, by Isaac Asimov (1967). This book for kids got many future Byzantinists started on their path. It's out of print, but you can get it used.