Hey Guys and Gals,
I created an account just to respond to this thread. This is a recent thing for me so it is fresh in my mind and I think that I can help. True All-Star mode on hard seems a bit daunting at first. I recently just started playing again (I took a hiatus.) I wanted to practice with Mario so I hit up TAS on Hard and got creamed. For a few days even. Then the old rhythm hit me.
The real trick to All-Star isn't actually fighting. It can be done with any character and in almost identical ways. Once I remembered what I was doing and beat it with Mario, I went through most of the upper list (Luigi, Peach, Bowser, etc.) one right after the other without a problem. As I was saying, the trick is to not fight as much as possible. It is all about the stage you are on. Always get in a corner and on an upper platform. From there, literally spam your USmash and SSmash attacks. Be sure to dodge occasionally if you miss. You can easily clear the first three stage and only take 50% or less damage. After that you're free to use whatever healing items you want without needing to conserve. The trick obviously is a bit more difficult on stages like Pilot Wings, but on those just stay near the edge.
You truly won't need any skill or practice with a particular character to do this. You don't need to know their strengths and weaknesses. It works! If I could post a video I would. That is all you need to do and it works with any character regardless of how strong or weak their attacks are. Three or four good smash attacks and the enemy will be flying. Many times it doesn't even take that. After you get a rhythm of doing that, you can clear All-Star under just a few minutes making this challenge a breeze (even boring at times.)
P.S. If you happen to end up on Duck Hunt stage in round 5 or 6, do not use the tall tree on the left. Instead, go to the short one on the right. Being that high and with all of the platforms underneath you makes you an easy target for a lucky smash hit from the CPU.
Edit : And for those of you who lose track (like me) of who you beat it with, True All-Star on Hard will always give you over 2 million in High Score for a character. Anything less than that and you will probably have to do it again (usually means you made it to the last stage, but didn't quite finish.)
tl:dr : Immediately get on a platform on either side of the stage and spam USmash and FSmash.