Oh, sorry. The question is really simple:
Do I need a wallet to start mining and "keep" this bitcoins?
Can I just stay with my bitcoins in my computer?
The wallet doesn't have to be on the same computer. For example, you can mine in a pool and the address to receive bitcoins could be from any wallet located anywhere.
You don't need a wallet if you are mining at a pool. The coins are kept at your account in their website and you can use them from there. If you want to transfer them to your computer, then yes, you need to download the bitcoin client which automatically creates a wallet for you.
If you are solo mining (which is not recommended unless you have lots of GPUs) you need to have the bitcoin client running, therefore you will also have a wallet.
You can mine, use and trade bitcoins without ever installing the bitcoin client because there are lots of websites/exchanges where you can keep your coins. If you don't want to trust your money to a third party, then make sure that the computer where you store your wallet is clean from virus/spyware.
A wallet is something you'll usually keep on your computer, it is not an external website.
A miner will generate a fresh wallet on startup (which you should backup immediately and perhaps regularilly). The location of the file on disk varies between OSes. Then, when the miner is run, this wallet will start filling up with coins (over time. If you mine Bitcoin, your throughput will be low right now).
If you're new to mining, make sure to read about Mining Pools.
Thank you for your interest in this question.
Because it has attracted low-quality or spam answers that had to be removed, posting an answer now requires 10 reputation on this site (the association bonus does not count).
Would you like to answer one of these unanswered questions instead?