I tried this on VirtualBox with Legacy BIOS/MBR. You can try to use UEFI/GPT if you want it.
You will need a Linux Live Media (Live CD or Live DVD) or Linux Rescue System (every Linux installation media has it, I believe). You will need it twice, before Remix OS installation and after Remix OS installation, just before Remix OS first boot.
First, you have to boot to Linux Live Media/Linux Recue System. Create a partition with EXT4 filesystem (Windows can't do this, this is why you need Linux). Since you want to install Remix OS as the only operating system, you can make a partition with all available free space on your hard drive.
After you create a EXT4 partition, reboot your PC, then boot to Remix OS disk. At GRUB screen insert 'INSTALL=1' in boot option, this will head you to Remix OS installation procedure. In partition choice, you can choose to format with EXT4 (again) or not (since you already formatted it). In system read/write option, choose no, so, you will be able to get OTA update in future. In GRUB install, hit 'yes'. In GRUB EFI install, hit no (if your system is UEFI/GPT, then you should choose no to install GRUB and choose yes to install GRUB EFI, you know it). Then follow the installation until finished.
After finished installation, reboot your PC again to Linux Live Media or Linux Rescue System, don't boot to your installed Remix OS. In Linux Live Media/Linux Rescue System, open/mount your Remix OS partition, you will find 2 main folder/directory, 'android-YYYY-MM-DD' (YYYY = 4 digit years, MM = 2 digit months and DD = 2 digit dates) and 'grub' folder/directory. You will see another folder/directory called 'lost+found', but that is part of EXT4 filesystem, not Remix OS it self. Rename 'android-YYYY-MM-DD' folder/directory to 'RemixOS'. Then open 'grub' folder/directory, open 'menu.lst' file, edit all 'android-YYYY-MM-DD' strings to 'RemixOS', delete 'CREATE_DATA_IMG=1' string, then add 'DATA=/RemixOS/data/ (this will make Remix OS use all available free space of your EXT4 partition into your Remix OS data). You can edit grub timeout in menu.lst too to 1 or even 0, so you don't have to wail too long on every Remix OS boot. After you edit all necessary strings, save that file. Then reboot to your installed Remix OS.
I hope that is good enough to understand. :D