To start my solution, I want to say that I know this can glitch on you and fail to reward you with the achievement after Shang Tsung.
This happened to a friend of mine, but it did not happen to me. I got my achievement the very first time after beating Shang Tsung.
The only thing I can tell that we did differently, is that when continuing after dying, I ended up trying all 7 characters throughout my ladder to Shang Tsung. My friend did not try every character. So as long as you have chosen to continue and selected each character at least once, then you should get the achievement. UPDATE: My friend used every character and even won a match with each character on the way to Shang Tsun and still did not unlock the achievement. I believe the only other way to try to unlock it if the standard first playthrough doesnt work is to beat MK1 with each character individually. Im not 100% sure.
On to actually beating the game, the AI can be brutal.
The best way to beat the AI is to use what I like to call 'exploitables'.
One way that may work on an opponent is to wait for them to come close to and then repeatedly tap both punch buttons (high punch and low punch) individually, not together. This series of punches is effective and will drain their health quickly.
Endurance matches can be a pain as you get no renumeration of health when you beat the first of two opponents, and you must win 2 rounds.
Another exploitable you can use is to repeatedly jump towards and air-kick them just as they get up from the ground. You can induce a rhythm where the AI will refuse to block your airkicks many times over giving you an easy victory.
When the AI gets close, and it will seeing as though they love to rush you and pin you to a corner, the best move to use the back+low kick slip move. You can also land an uppercut if you time it right to get out of a situation.
For Goro, the best technique to use is to try out different fighters and keep air kicking towards him, then air kick away from him. Its important you kick when jumping away from him as that will help prevent him from rushing you while you land back in preparation for your next air kick. I found it easier to use Liu Kang as he seems to jump higher than others, and his air kick lands in a higher position than others' kicks, giving you a better chance and jumping back if you hit him and he blocks.
Shang Tsung is incredibly difficult to beat.
The only way I managed to beat him was with Scorpion is to duck while he was in grandad form. Keep ducking to avoid his skulls, dont even try and block them as they deal heavy damage.
Then when he transforms into a fighter, lure him close to you and use the repeated punch combo exploitable on his fighter form and fingers crossed you will have dealt enough damage before the round is too late to salvage. Scorpions down+back+high punch move can also help as it is a quick teleport punch.
Dont even bother using any projectiles or fancy moves as it seems the AI was programmed to read your mind and counter almost everyone perfectly, especially in the later fights.
All these tips come from me beating it on Normal, so they may be more effective on Easy if you change the difficulty.