This question came up a few minutes ago as I noticed I have a lot of first hand sources for the history of other famous leaders - for example the complete works of Josephus , the works of Lucy Mack Smith, a book of first hand accounts from the American Revolution etc - but nothing on the history of Anne boleyn and .Henry VIII.
Boleyn is a fascinating figure for me but one who also feels a bit inaccessible. Whilst I have read some good books on her - most notably The Creation of Anne Boleyn - I have yet to find any first hand sources. The same goes for her husband Henry VIII.
What accounts and sources would you historians recommend?
There is an enormous wealth of material published on the reign of Henry VIII, and much of it is available for free online.
If you are looking for narrative sources, the best would be:
Ellis, ed. Hall's Chronicle; Containing the History of England...to the End of the Reign of Henry the Eighth (1809) This has other versions and titles, but this is considered the standard work, and is available here.
Polydore Vergil, The Anglica Historia of Polydore Vergil, A.D. 1485-1537 edited by Denys Hay, 1950. The majority of the information stops around 1530, but it is an indispensable source for the beginning of Henry's reign.
On Anne Boleyn specifically, there is a short work edited by Maria Dowling in the Camden Miscellany Volume 30 (Camden Fourth Series, Volume 39) entitled "William Latymer’s Life of Anne Boleyn", which may be useful. There are other works that mention their relationship but are more focused on the Reformation in England at large, so I haven't included them here.
Two sixteenth century biographies are also excellent resources for the reign of Henry - published together in 1962, they are Two Early Tudor Lives: The Life & Death of Cardinal Wolsey by George Cavendish and The Life of Sir Thomas More by William Roper.
The letters exchanged between Henry and Anne are printed in Savage, ed., Love letters of Henry VIII. More letters of Henry are printed in Byrne, ed. The Letters of King Henry VIII (1935).
There are two main chronicles for the period as well. They are Hamilton, ed. Wriothesley's Chronicle; A Chronicle of England During the Reigns of the Tudors, and Nichols, ed. Chronicle of Calais in the Reigns of Henry VII and Henry VIII.
The vast majority of documents for Henry's reign aren't narrative sources. They are the printed letters, documents, and summaries compiled in various collections. The largest by far are the 21 volumes and 2 addenda (in 48 parts) of the Letters and Papers, Foreign and Domestic, of the Reign of Henry VIII (L&P). All the volumes are free online, either through archive.org, google books, or British History Online (www.british-history.ac.uk). The vast majority are dry documents, but they are the standard works.
More expanded versions of documents found in the L&P are printed in State Papers of the Reign of Henry VIII, 11 volumes (divided by chronology and subject matter, often containing more information than the L&P; Calendar of State Papers, Spanish Volumes 2-8 (Containing the very informative letters in translation of Eustace Chapuys, the Imperial ambassador in England); Calendar of State Papers, Venice, Volumes 2-5; and Byrne, ed. The Lisle Letters 6 volumes (Correspondence of the illegitimate son of Edward IV, and Lord Deputy of Calais).
If you are going to be interested in a wife of Henry VIII, Anne Boleyn is a good one to be interested in because there has been a lot of work done on her. As someone studying Catherine of Aragon (who often gets overshadowed by Anne), I am very jealous.
That being said, you need to be careful of what you take as "gospel truth." The mythos surrounding Anne Boleyn is a whole beast of its own. Tangentially related, Palgrave has a great series called "Queenship and Power" (http://www.palgrave.com/gb/series/14523) that covers queenship and power. One of the books is History, Fiction, and the Tudors (http://www.palgrave.com/gb/book/9781137438812) which might be interesting.
Other books in this series with chapters I liked that also have chapters covering Anne Boleyn which I did not read are:
The reason I include these books, other than them being awesome, is that they tend to reference primary sources you might find interesting. Although reading through thousands of pages of catalogs can be super fun, reading books like these and then looking at their sources can be a nice shortcut.
Back to primary sources. Elizabeth Norton (http://www.elizabethnorton.co.uk/) has published a book called Anne Boleyn: In Her Own Words & the Words of Those Who Knew Her. I have not read it but I have read other things Norton has written and liked them. She also published a book called the Anne Boleyn Papers, no idea what the difference between the two is.
Norton's degrees are in Anthropology but I don't think that excludes her from also doing good work in history.
One thing I want to mention about the British History Online. I have found the website somewhat terrible and prefer to find the catalogs on archive.org instead. While the online website is searchable, things can be missed and if you are going to go full in, scanning the catalogs (and not just relying on the index) is the only way to be sure. A lot of the catalogs are from the Victorian Era (seriously) and I've found women to be overlooked in indexing.
Mary Anne Everett Wood Green, who is the editor of some of these catalogs, published a great two-volume series called Letters of Royal and Illustrious Ladies which has some letters of Anne Boleyn in it. Being published in 1846, you can find both volumes on Archive.org!
Anne Crawford's Letters of the Queen's of England also has a few Anne Boleyn letters in it. There may be some overlap with the Mary Anne Everett Wood Green books, although there were a few new letters of Catherine of Aragon in Crawford's book.
There is also the Intimate Letters of England's Queens by Margaret Sanders which is somewhat salacious looking. I have not read it and do not know what it is like.
A final word of warning, there are letters that may or may not really be Anne's. One case is her alleged final letter from the Tower to Henry VIII. Make sure to take things with a grain of salt and read up on which letters are debatable. Things like that might be covered in the Queenship and Power series, I have no idea.
Finally, HAVE FUN! I think it is so important to look at these women in their own words instead of the words of the men in their lives. Although not everyone left a big paper trail, there are letters that deserve to see the light of day!