The problem you are having is specific to your system. There are many people that have installed the 6.1 update without any problem.
So without more details on your system it will be hard to give you suggestions as to how to overcome the problem you are having.
I am running Windows 7, 64-bit.
Photoshop CS6 (not CC) is installed.
Update to LR6.0.1 worked fine.
It is a Dell Desktop with Intel i7 CPU.
Sorry I don't understand your post. This is the Lightroom forum and your original post states Lightroom 6.1 hangs at splash screen. Then in your last post you mention Photoshop CS6.
Only thin I can suggest is to delete the Preview Cache folder. It could be hanging because of a corrupted Preview cache.
If that fails to correct the problem Uninstall LR and reinstall the 6.1 update file which is the complete program from this link
Thanks for your help Shootistbond007.
I mentioned Photoshop CS6 because its presence may interfere with LR (which - of course - it should not!).
I do not believe that the Preview Cache Folder causes this issue. Reason: it even happens with an empty database.
Your link does not contain any reference to LR6.1 - or at least I cannot find it. The highest version under this link is 5.7.1.
Both LR CC and 6 are the same download. It is how you register it that counts. Both are registered by your Adobe ID email and password. If you have a subscription it becomes the CC version. If your Adobe ID doesn't have a subscription then you are asked to enter a serial number.
Just like it was when you first installed the stand alone LR 6.