I'm trying to write a biography on her and I'd like to know what sources and books can I use for it.
From a quick search, it looks as if there's a handful of sources on her directly:
Barrett, Anthony A. Livia: First Lady of Imperial Rome. (Yale University Press: 2004).
Bartman, Elizabeth. Portraits of Livia: Imaging the Imperial Woman in Augustan Rome. (Cambridge University Press: 1998).
Dennison, Matthew. Livia, Empress of Rome: A Biography. (St. Martin's Press: 2011).
The second one is more focused on artwork depicting her, but according to a review from Susan Wood in American Journal of Archaeology,^1 it sounds as though it provides an analysis of the shift in the role Livia played in her life and after, and how imagery depicting her (sculptures, etc.) changed accordingly. Definitely not something to ignore in a biographical work.
Apart from that, your best bet is likely to focus on sources discussing related subjects, such as Augustus, the Roman emperors, women in Roman society, etc. For example:
Potter, David. Emperors of Rome. (Quercus: 2008).
Fraschetti, Augusto. Roman Women. (University of Chicago Press: 1999).
Also, you could find online biographies and follow their sources. For example, this biography from an old website on Roman Emperors looks to have a handful of sources you could follow. Wikipedia and Biography.com will often also have some good sources to follow up on in the bibliographies of their respective articles.
Note: This is my first time commenting in this sub, so I'm not sure how rule #4 would apply to this kind of question as opposed to the standard kind of question this sub gets. Tried to follow to the best of my interpretation.
Original sources or more modern works?
For ancient sources you’ve got the usual suspects: Cassius Dio, Tacitus, Suetonius.
The last modern work on her I can recall reading is Dennison’s Livia, Empress of Rome: A Biography which I quite enjoyed.
Disclaimer: not a historian, just an ex-classical history minor.