Well unless you tell your opponent and you both agree to use a computer. Else that is considered cheating if your opponent uses a computer.
My opponent uses a computer

It is against the rules for rated games, which he seems to be using it for.
Of course, that is if he really is using a computer.

If you want to report this case to the staff, there is a little "report abuse" link at the bottom of each page.

I already have, of course. Right now I'm so upset I'll have to await making any moves in the rest of my games for a while. lol

It is against the rules to use a computer engine - however, it is not against the rules to use computer software that include tables or databases. However, I understand that the implication from his profile is that he is using a computer engine and if that is the case he would almost certainly be banned. You have done the right thing to report it.
However, just because someone says they are using a computer doesn't mean that they are any more than someone saying they are not using a computer being proof that they are not. Leave it to the admin staff to ascertain whether he is or isn't.

what does he get out of playing with a computer? i dont understand all this,if your using it to play why bother playing, why doesn't he just put two computers together, let them play,and he can just watch?

I believe he is using a computer for rated games. Just take a look at his rating - 2320 and rising ...

what does he get out of playing with a computer? i dont understand all this,if your using it to play why bother playing, why doesn't he just put two computers together, let them play,and he can just watch?
lol. I agree! But, I guess it's something like stealing candy from a child: there is always someone getting a kick out of "outsmarting" someone else, no matter who they are.
It's a joke: I've even complimented this guy for his game, and he didn't even then reveal his methods. He probably gets a kick out of stealing candy from kids as well ...

at least he tells, you, but it seems dodgy, he will never blunder,
He never told me, though he did leave a trail for me to find out myself. But, meeting him in a tournament: what could I do?

He is not allowed to play "advanced chess" , that is using a chess engine to help him with analysis, in rated games whether or not his opponent agrees. The rules are very clear:
You many only have ONE Chess.com member account. You may NOT get any help from any person or any chess engine throughout the course of a game, including tablebases. You MAY use books, magazines, or other articles. You may also use computer databases (including Chess.com's Game Explorer). EXCEPTION: If both players agree for the use of a chess engine in an UNRATED game then it can be allowed.

He is not allowed to play "advanced chess" , that is using a chess engine to help him with analysis, in rated games whether or not his opponent agrees. The rules are very clear:
You many only have ONE Chess.com member account. You may NOT get any help from any person or any chess engine throughout the course of a game, including tablebases. You MAY use books, magazines, or other articles. You may also use computer databases (including Chess.com's Game Explorer). EXCEPTION: If both players agree for the use of a chess engine in an UNRATED game then it can be allowed.
Thanks, for your support, MM78!

well i think people who use computers during games are gits, chess is about two humans using tactics to outwit each other,apart from white moving 1st its a level playing field. why should fair chess players have to put up with this? i work hard for every point of my rating and i'd consider leaving the site if i was to have to keep facing cheats who prop up their poor game with outside aids. i know its hard to proove,as many have stated above,but the idea of it makes my blood boil. (rant over)

Thanks for your support CATLOCK! I've worked hard for all the rating I've got as well. I refuse playing anyone using a computer, (thats why I never challenge any very high rated players) but what to do when you meet them in a tournament?
I really hope the people in charge of this site takes this matter serious, and does something about it. I've recieved a message from a member of the staff denying me to write any of his opponents about this matter (because I did send the first ones on his list a message, thinking it was the only right thing to do, so none of them would resign or lose a game against him before this matter was settled) but as long as I've not been denied writing in any forum about this, I'll continue to do so.
If this is a site that supports honnest chessplayers, I expect a member of the straff to make a comment in this forum, and not just sending me a message telling me to be quiet about it.
I have turned folks in before and have seen those accounts closed. They staff will look into any report of this nature and take the steps needed.

I have turned folks in before and have seen those accounts closed. They staff will look into any report of this nature and take the steps needed.
What happened to the games the cheaters had already won?
The games that had been completed stand as is. It is impossible to go back and fix all the issues to completely unwind all that happens when someone cheats. For instance, suppose my losing a game to a cheater kept my rating low enough to enter a tourney I would not have qualified for had I not lost.
In the Season Match - Round 1 - Unity Group vs Chess Unlimited - Board 7, I'm playing a guy with this profile:
Note that I am playing so called "Advanced Chess" (http://en.wikipedia.org/wiki/Advanced_Chess). If you are not happy with this, please don't play with me.
How could I avoid playing him when he has joined a tournament? No need to say: I feel this is really unfair, and I wonder if using a computer is according to the rules at chess.com.