If you are asking if you can solo mine with the goal of getting a transaction confirmed quicker then yes you can do that however this would really be a moot point unless you are trying to broadcast a transaction with either a 'lot' of inputs or a 'lot' of outputs and are not paying a sufficient fee.
If you are asking if you can mine so that the block reward goes to 'your' address, then this is what every pool (except eligius) does as the block reward goes to the pool's address and then the pool will in turn pay the pool's users. (or someone who is solo mining will simply have the block reward go to their own address)
I have a address that revives and sends out hundreds and hundreds of tx everyday, some of them are so small that the fee needs to be so low that it really don't matter. That's why I thought it would be a good idea to point some of my miners towards some specific transactions to make them confirm faster.
If you have miners running they are most likely mining for a pool of miners. Then it's up to the pool operator to decide what transactions to confirm. Your miners are only doing what the pool operator tells them.