Freeze problems - the spinning beachball strikes again

Sadly this means nothing to me as I'm not technologically adept! Previously I've been limited to what backup points I could use from my disk; if I tried the latest, June 26th, it seemed to restore but then I couldn't get the log-on screen. I just tried June 12th and it allowed me in, but I got a strange message about the Time Machine disk having changed, and how I could try to use it or that it could be someone trying to trick me into using the wrong disk. When I do manage to log on, the behaviour of the external hard drive is very odd - when I enter Time Machine the backups seem incomplete, and some items aren't available for restore. Also, the disk seems to take forever 'preparing backup' and then, as I said earlier, tends to freeze on 94mb, and the whole rigmarole starts again. Time Machine preferences say 503 gb of 1 tb available, but oldest backup - 'none'

I've tried safe boot but the progress bar stops about a third of the way in and the spinning gear comes back. I've tried all the 'how to get past the grey screen' hints. Someone has suggested I try a clean install because I may be restoring the cause of the problem when I restore the system Trouble is, I'm not sure how to do this. Any clues? oh,and my Time Machine is whirring away now, stuck on 9.16gb of 311gb