I am addressing some key points in your post to try and minimize your concern with this situation. I am assuming you actually have proof of him doing this and that you didn't just imagine his complaints about you behind closed doors out of fear and paranoia. I am also assuming that you are indeed not a total screw-up and that all the other bosses like you.
I work in a matrix-management based company where multiple superiors decide my performance appraisal, salary increases, etc. I have no single manager.
I am aware that he is making loads of complaints to his boss and I fear for my job as he is one of those who can decide where my career goes next.
He is one of several bosses that have a say in the future of your career. If I were the boss of bosses and the others had nothing but positive things to say about you but one of them was "constantly complaining" about you then I wouldn't just assume you were a bad employee. I would assume that this one guy just doesn't get along with you. If he doesn't get along with you then he is probably highly critical of many talented hard working people so I won't take what he says very seriously.
Furthermore I would look down on somebody who is constantly complaining about his subordinates and blaming them for his project failures. He is making himself look like a bad leader to his peers and his boss.
but the one occasion I hadn't double checked he had intentionally left things for me to correct - without telling me. He then told me it was "obvious" that I had to check.
That is a serious allegation. Think about that. You are accusing him of purposely delivering shoddy work with the sole intent to try and catch you in a trap. Incredible allegations require incredible evidence. Now I don't believe you and I barely know you so it is best not to bring this up even if you suspect it. When you talk to his boss make sure that you exhibit positive and strong personality traits.
Another example, he sent me some stuff to distribute/communicate across the company. I am not responsible for double checking though I sometimes do out of my own initiative -
- Do not talk bad about anybody in the workplace. You always want to be seen as the guy who gets along with everybody.
- When you make a mistake then own it and display humility. You actually look stronger when you admit your mistakes.
- When you admit a mistake, always follow it up with how you will try to make sure that doesn't happen again, a plan helps.
- Never blame anybody. Give the facts without emotions. You were given a work from the one boss and he expects that we should always proofread for mistakes. I should have remembered this but I did not. I will make it a note to always proofread his documents in the future. Leaders don't care about blame, they care about figuring out the solution.
Now you leave it up to the bosses boss to decide if proofreading should always be a part of your responsibilities in the future or if the problem boss was wrong to expect that of you.
For example, there were a few times where he set expectations to his superiors that he would be able to deliver something, based on the assumption that I would have gone above and beyond the day before and prepared that - however, he hadn't asked me. He only reprimanded me after he failed to deliver to his boss.
And his boss will reprimand him for not delivering when he promised. Again he looks even less credible and trustworthy now to his own boss, especially if he starts blaming his subordinates. It is pathetic really. His reprimanding you is shallow in comparison to the damage he did to his own reputation.
Performance appraisals are in less than a month... so what can I do now? I am thinking of telling this superiors' superior in order to, well, protect myself. Is that a good option?
If I were you I would ask the bosses boss to have a quick chat but don't talk about your problems with the boss. Just point out that you have made a few mistakes here and there, give some brief details and facts about what your boss asked you to do and where you dropped the ball, but that you really wanted him to know that you are working on it, and that you won't let the company down.
You will look like strong and likable and your trouble boss will look untrustworthy and like a poor leader.