Hi Charmy! If you feel your son is playing too much video games (I am assuming this is an x-box or computer game?), you can set time limits for each day. Those types of games can get very addicting. I would set aside a certain time each day that ends about 1/2 hour before bed time. Make it for about 1 hour max (we know how fast time goes when playing a video game). Set the time for about 10 minutes before time is up so that your son can finish whatever he is doing.
When you state he crys over the game, it reminds me of when my son was that age (11 years old) and would get so frustrated because he could not accomplish a certain level of a game. You state that you give him permission to play this game leading me to believe that there may be controversy over kids his age playing it? If it is marketed to an older age, it really may be too hard for him even though he enjoys it. I remember those days and my comments would be "If it makes you so upset, then don't play it!" Of course, that comment didn't go anywhere with him - he would be angry with the program and work that much harder to overcome it and win. So, I am assuming that the grounding was not so much that he cried but more so that instead of admitting that he was in over his head, he kept at it thus getting more and more upset making for a very ugly scene at home where you are telling him to stop and he is bound and determine to conquer this before stopping.
I think I would sit down with him when he is calm and before he plays again and set up some ground rules. Explain to him that the game was built for older kids and there may likely be parts of it that he is just not ready to understand. Ask him to try to be aware of his frustration level and ask himself if being angry is really worth continuing that level. Maybe taking a break for a day and coming back rested may help or maybe it is just something he has to wait until he is older to do? Can you find a game that has similar goals but more toward his age group?
Hearing your kid say he hates his life can be scary. I think it is common for kids these days to express that but as a parent, only you can pick up on how serious that is. Many times kids do not even think about ending their life if they hate it but then again, there are kids that do. Being 11 is very hard. Definetly too old for little kids interests but also too young for older kids activities. Too young for the most part to leave alone at a movie theator or bowling alley or local skating rink (depending on the kid and where you live). So still needing adult supervision but feeling like they should be able to do more grown up things. If his comments really do scare you, talk to his doctor.
Getting him involved with an activity with other kids would be very helpful. Is there a sport he is interested in? A reading club at the library? Does he have siblings who will play board games with him? I bet he would love to play a board game with you. That may even become a way to get him off the computer or x-box or whatever he is playing on each night - once he is signed off he gets to challenge you to a board game or card game.
So, in a nut shell, set up limits so that he knows every day how much time he has, talk to him about his reactions to a difficult level on the game, and when you do take this time away, find something positive to fill it with.