5.0 is the most recent official software for the Note 3. If you want anything newer than that you'll have to root it and install a custom ROM based on a newer version of Android (a "custom ROM" being a hobbyist build of Android for the particular device).
If you want to go that way you need to do your research before taking any actions. The place to start is the
XDA-developers forum. If you follow that link you'll see the first problem: there are many versions of the Note 3, and ROMs developed for one version will be incompatible with another. So identify the version you have and go to that area. You are looking for forums (for the correct model) with "original development" in the title: in XDA-speak "development" means "ROMs based on manufacturer's software" while "original development" means ROMs based on the Android Open Source Project (AOSP). Since you want newer than Samsung have released then ROMs based on Samsung software won't help. Depending on which version you have you might find ROMs based on Android 7, or you may find no ROMs at all. If it's the latter than it's easy, there's nothing you can do. If the former you'll then have do read about the ROMs to decide which one suits you best, then read how to root the device (here the "development" forum for your model might be useful) and install it. Note that some of the tools used for rooting will also depend on the model you have. I can't advise because I don't own this phone and never did.
Final warnings: if you like Samsung software features then an AOSP ROM won't have them, it's not Samsung software. Read what the ROM supports and what it doesn't. And with a 10 year old device you might find that some files or tools are not available: it costs money for developers to host their software, so eventually the links stop working.
Also before installing a ROM make sure you do a backup of your previous ROM (the "custom recovery" you use to install a ROM can do this) in case you don't like it. And back up your data, as all of this switching android versions will require factory resets at times.
Sorry this isn't the simple answer you were looking for, but this is the way things are.