[Ethereum] All the mined coins are going to address 0x000000

If you are on a private net/chain you actually don't need to mine to get ethers, but just to confirm transactions.

You can fund your account(s) by specifying the amount in wei you want that account to start with, in your genesis json file:

"alloc": {
    "<your account address e.g. 0xaaabd38c8f1a188a0b8bbf93bdca420cfdd760aa>": {
        "balance": "10000000000000000000"
    }
}

Another thing you can do if you still want to mine to get ethers is lower the difficulty, again in the genesis block, a value of "0x400" or `"0x200" will let you mine faster the first blocks. If your geth is stuck and doesn't mine anymore (on osx I found it common when you put the computer in sleep), simply kill it with Crtl-C and restart it.

To specify your genesis block you can pass these parameters to the geth command:

$ geth --genesis <genesis json file path> --datadir <some path to an empty folder> 

A sample genesis block is:

{
  "nonce": "0x000000000000002a",
  "timestamp": "0x0",
  "parentHash": "0x0000000000000000000000000000000000000000000000000000000000000000",
  "extraData": "0x0",
  "gasLimit": "0x8000000",
  "difficulty": "0x400",
  "mixhash":  "0x0000000000000000000000000000000000000000000000000000000000000000",
  "coinbase": "0x0000000000000000000000000000000000000000",
  "alloc": {
  }
}

You can find a more complete guide here

You should calculate your average block mining time on something like etherscan mining calculator before trying to mine.

 It will take you an average of 117.39 days to find 1 Block

But mining difficulty will raise so much during this time that you probably won't mine anything. Pure lost plus you burn your hardware.

So unless you have at least 50MH/s, you'd better buy ether from an exchange and even with 50MH/s you should consider pool mining instead of solo mining.

Edit: this answer was written a long time ago and hashrate should be higher today to expect mining something. Also note that pools won't give you more chances to mine a block, it will only smoothen the rewards. Finally, today, maybe you should wait for PoS if you want to participate in securing Ethereum.