With the new hard drive the laptop won't boot at all unless you clone the old hard drive to the new one, and for that you'll either need a USB adapter or a mate with a desktop PC. Some laptops have the space for two hard drives, but they're normally reserved for the huge gaming laptops that you're unlikely to have. If you've got a mate with a desktop PC and the two hard drives are SATA, then you can just plug them into the desktop alongside his existing drives and they should work fine - laptop and desktop SATA is exactly the same. Make sure it still boots from his drive, and then see below for some software to use to clone your drives.
For imaging software, if you've got a Western Digital drive you can get a slightly cut-down version of
Acronis True Image Home Edition for free. If not, then something like
DriveImage XML should do the job for free, though while I've seen it recommended, I don't think I've ever used it myself.
For the record, if the old hard drive starts misbehaving and refuses to be recognised, then I find the controller on USB enclosures to be more resilient and more likely to detect a dying hard drive than the controller built into the laptop. That said, if the hard drive is dying but is still detected, then it's more likely to be developing bad sectors rather than an electrical fault, so that trick would be unlikely to work.
Two other tricks for a dying hard drive to use as a last resort - take the drive out of the laptop, stick it in a freezer back, seal the freezer bag and throw the hard drive in the freezer for an hour. If this works, you may need to repeat the trick a few times in order to get all the data off it, as the drive will start failing again as it heats up.
That one's safe so long as the freezer bag is tied tightly. This one is more of a last resort. Take the drive out of the laptop and whack it off a desk/bench/wall. Sometimes it's enough to free a jammed head. It's worked for me once, but it's only to be used as a last resort, and obviously don't do this if you intend to send a drive off to a data recovery firm...