Is your kernel loading the module/driver for your AMD/ATI Radeon 7870?
<OR> did you install a proprietary driver?
Post the output of your /etc/X11/xorg.conf.d file and maybe someone will see what's wrong; if anything.
A member here that is in Debian Testing told me this:
***If a system won't boot to desktop it has to be xorg, the kernel or maybe a proprietary video driver most likely.
It could also be something in the init system (sysvinit, startup and systemd) but in testing it's xorg related.
Run a update, if that doesn't work try Recovery Mode and direct it to boot when it get's around to asking what you want.
Recovery Mode runs a few more corrective tools and may fix it on it's own.
If you can run the DE as root the problems is with your user not the system at large.