First you'll need to prep the HD before you put it in. Using an external FireWire case to hold the drive would be best (USB is just so slow). In any case use the Disk Utility if you have something older than Mountain Lion if you have something newer use the OS installer to prep the drive as it will create the recovery partition.
Once the drive is prepped and has the OS on it use the Option key during the restart to select it to see if it starts up. If it does using the new drive externally you know the drive is ready.
OK, now the question on copying over your stuff. Using a clone application you can copy over your stuff. If you have a second Mac you can use Apples Migration Assistant. These will both image the drive with everything you have over. The risk here it will also carry over of the hidden junk.
A TimeMachine backup is typically journaled, which basically means it is a stepped backup so you have not only the core backup you also have all the old stuff as well. I wouldn't recommend using your current backup. Instead I would create a fresh one using a different HD so it is a snapshot of what you have now without the weight of all of the changes.
I do recommend you first do a good housecleaning getting rid of the stuff that is not needed. In addition you should think about getting a log and cache cleaner app to flush them out as well. Then I would repair the your current drive repairing the permissions and the drive structure so the clone or backup will be cleaner.