Macbeth actually reigned from 1040 to 1057 (Elizabeth I died in 1603). He did become king after Duncan's death, but the circumstances were quite different than Shakespeare's portrayal. Remember that Shakespeare was not writing history, but entertainment, and was especially concerned with making his royal patron, James I, happy.
I would recommend the link below for more information about the real Macbeth, his life and times. The history of Scotland is fascinating, so I would also recommend you check the library for further reading! Good luck!