Since you've already bought and installed a new hard drive, using a LiveCD of Linux or even a WinPE won't do you much good, since the hard drive you want to recovery files from isn't even in the laptop any more.
The problem is with how the drive has failed. If it won't spin up, or you can't read data off the drive, then your only option is an expensive one, since the drive must be sent to a data recovery center where it can be taken apart in a "Clean Room" and the platens put into a different chassis. That isn't something you can do reliably on your own.
If the drive still spins up, you MIGHT be able to recover data from the drive with a USB to Sata or USB to IDE converter... but it would stink if you spent $25 to $50 for one of these things, only to find out that the drive is dead.
So, here is what you *can* do that won't cost too much money. You didn't say whether your drive was an IDE or a SATA type drive... so here is one of each...
IDE http://www.newegg.com/Product/Product.aspx?Item=N82E16817145082
SATA http://www.newegg.com/Product/Product.aspx?Item=N82E16817388019
Take a look at those... and you can (and should) poke around your favorite online retailers for ones that are cheaper. Buy the one that is made for your type of drive, install your drive in it, and plug it into the laptop. If the mechanics of the drive are OK, you should be able to retrieve your Data. you *might* have to run CHKDSK on it in case the data has been corrupted. But be prepared for not being able to get any data from the drive if the drive is dead. Not unless you want to ship the drive off to a Recovery Service and pay between $400 and $1000 for the job.
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