I have read that there were issues running it on an application called Bluechips - which is typically what people use to run Android apps on PC. It essentially lets you log into the play store under your own account, and emulates itself as a Samsung Galaxy S2 variant I believe - may be totally wrong on the phone type.
Unsure if this has been rectified as Bluechips does have a fairly aggressive development schedule.
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If you don't have the Bluestacks beta, try it instead as it lets you use Google Play. Make sure you're logged into Google Play so the game will verify and retry the download. The failure to download is usually a license check failure.
I made a forum post previously about Bluestacks and FFD not working together. I have left a few threads on Bluestacks "official" support forums, that were all ignored with no responses or emails or anything.
I have sent several "reports" via Bluestacks' program itself and ever time I uninstalled Bluestacks I selected the option that an app wasn't working, and listed Final Fantasy Dimensions. There hasn't been an update since I tried, or even further, no more responses.
Windowsandroid apparently isn't ready for this game yet, but it's extremely promising. With some tinkering, I got google play going and spoofed it to think I was using a different device, but it still lists all apps as "incompatible with your device". In the future, I imagine this will be a viable way to play FFD on PC.