I have made a comparison of the anime and manga of Akame ga KILL! and included suggestion on where to pick up the manga if you know the series by watching the anime.
Basically, if you don't want to read too much overlapping content, start reading from chapter 39. The previous chapter (chapter 38) covers the fight between Mine and Seryu, which corresponds to episode 19.
There are many differences between the anime and manga from that point on. The Wild Hunt arc (from chapter 44 to chapter 48) in the manga introduces several new characters, some of which play important roles in the overall storyline beyond their arc.
Since the anime skips this arc, the fates of several characters were changed:
- Run was severely injured in chapter 48 after he defeated Champ, who was the perpetrator behind the massacre of Run's students back in his village. Before Run died, Kurome stabbed him with Yatsufusa and made him her puppet. He was alive and well at the end of the anime, though.
- Bols' wife and daughter was brutally murdered by Wild Hunt for their amusement in chapter 46. They stayed alive at the end of the anime and were seen distributing ration.
- In the manga, Suzuka - one of the Four Rakshasa Demons - actually survived from Tatsumi's attempt to crush her by destroying the whole building. She is assumed dead in the anime.
Therefore, although you might encounter some similar scenes in some of the chapters, it is best if you read from chapter 39 to understand the context in later chapters.
Apart from the main chapters, there are several extra chapters which are not adapted in the anime:
- Volume 5 - (Unnamed Special Chapter)
- Volume 8 - Extra Chapter (33.5) "Kill the Reminiscences" (追憶を斬る Tsuioku o Kiru)
- Extra Chapter (7.5). "Kill the Blackness" (暗黒を斬る Ankoku o Kiru)
- Extra Chapter (6.5). "Kill the Mad Swordman" (剣鬼を斬る Kenki o Kiru)
And if you are interested, you might want to read Akame ga KILL! Zero, which tells the background story of several characters in the series.