Download Samsung Galaxy Tab 3 10.1 Marshmallow Android 6 ROM. An xda developer user has churned out a a custom ROM that lets you get latest version of android which is v6 marshmallow on the Galaxy Tab 3 10.1 tablet. The ROM is a no-frill ROM and is based on pure AOSP experience. You need at least TWRP 22.214.171.124 to flash this Galaxy Tab 3 10.1 Marshmallow Android 6 ROM.
* Copy ROM to SDCard
* Copy GApps to SDCard. I prefer opengapps.
* Copy SuperSU to SDCard, downloadable from here.
* Boot into custom recovery like TWRP.
* Backup EFS partition
* Wipe -> Advanced Wipe -> Dalvik Cache + System + Cache + Data
* Install ROM
* Install GApps
* Install SuperSU
* First start takes a long time
Almost everything is working on this Galaxy Tab 3 10.1 Marshmallow Android 6 ROM except for chromecast, houdini, RIL, MHL, secure videos like widevine, HW audio decoding and permissive SELinux.