I had an Asus pc with a McAfee LiveSafe installed on it. Also had a subscription which will expire at July 2017.
The pc was ruined and I bought a new Lenovo pc instead.
Anyway to remove the account from my old pc and use it on my new one?
You should be able to simply go to your account on your (New) Device/Computer and Download/Install your McAfee Software. As with an active subscription as it covers more than one device.
Of course you can always Contact Technical Support and Select/Ask for Customer Service and confirm that.
All the Best,
I might add that unlike on prior Operating Systems, Windows Defender is not compatible with your McAfee Software using Windows 10. As it contains Anti-Virus capabilities now. The McAfee installer should disable it as it is, however I thought I would mention that.