Minecraft wiki entry for Videos, associated text: If you plan on recording or streaming the game, we recommend OBS. It allows for both recording and streaming the game, as well as integrating a Twitch account to see live chat and stream options without using a browser. It's also free and can capture in any quality your computer can handle.
For video editing, you can use Vegas or iMovie. Adobe Premiere is also an option, with the Elements and Pro versions, depending on your budget. VirtualDub is a high quality open-source editor for those who do not wish to pay for an editing program. DaVinci Resolve is another popular, high quality editor, which has a free tier that is more than enough for editing Minecraft recordings.
If you wish to record your voice and the audio from Minecraft in two separate streams to be edited together later, you will need a program like Audacity. Or you can use the DXtory recording program which automatically records sounds in separated streams. Also, Blender can split the Sound and Video streams if you use RecordMyDesktop.