On Facebook I'm friends with a normal account (I think) that posts messages sent to it. I was reading a post of it that showed up on my news feed but when I went to like it I got a message saying this post was deleted. Is there anyway at all I can view this post, even if it's saved in my browser cache?
No, it is not possible to view this deleted post. Even if you have it in your cache, it would have already been overridden the next time you visit the same page (and you did, since you saw the message about the post being deleted).
There might be some luck if Facebook emails you regarding posts by your friends, as they may contain the message within the email so that you can see it even if it is removed from Facebook itself.