I experienced a very similar problem. You may be having a problem where Compiz is incompatible with your graphics driver. Compiz is not needed to get to the login screen so you get to that point OK. Then when you login, Compiz loads (or tries to) and blows up. There are lots of tips on this site regarding how to fix broken Compiz, none of which worked for me and several of which bricked the PC completely. A possible workaround (not the solution) is to try and load the legacy Unity 2D desktop rather than the newer Unity 3D desktop (which is a Compiz plugin and hence rides on top of Compiz. To do this, boot the computer so you see the login display. If you configured auto-login, you need to figure out how to disable this. I ended up completely repeating the install process from scratch and when prompted whether to auto-login or not I selected no. At the login screen, there is a Ubuntu icon immediately to the right of your login ID. Click this and you'll see an option for desktop options: Unity (the new Compiz-based version) and Unity 2D (the old Compiz-free version). Select Unity 2D and then click Back and then finish logging in. When you log into your desktop, lightDM will not try to load Compiz at all. If your desktop then loads fine, you at least have a functioning system and know where the problem is, even if you are missing a bit of bling.
Running the command
echo $DESKTOP_SESSION in a shell will tell you whether you are currently running Unity (3D/Compiz) or Unity 2D (no Compiz).
To make permanent your decision to use Unity 2D, run
gksu gedit /etc/lightdm/lightdm.conf and change
user-session=ubuntu-2d. Save, exit gedit and reboot. You should not have to manually select Unity 2D vs. Unity 3D at login as lightdm will read its config file and automatically load Unity 2D. Good luck.