Redmi Note 3 Has a Locked BL and Recovery is not accessible. You have to Flash Custom Recovery to Access Recovery(TWRP). You need to Root your device. Rooting your device needs Unlocked BL and PC. Rooting is impossible without PC.
You can't root your device without unlocking bootloader,only China developer rom comes with native root but in order to activate that you require unlocked bootloader.
Read and Follow the FAQ
Mi Devices Bootloader Unlocking FAQ: Announcement, Troubleshoot, and Tips
Tutorial for Successfully Unlocking Bootloader Stuck at 50% !
Here is a guide for unlocking: