Can I put a bigger hard drive in my Dell Latitude c600 laptop?

You need a laptop IDE hard disk.

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To install a different hard drive is not that hard. The biggest issue is making sure you have the restore disks for windows. Once the hard drive is changed, you will need to install Windows and put in the Windows activation code. Also make sure you have the drivers for you specific computer on a CD or USB drive.
The easier thing for most people is to buy an external hard drive and just plug it in via USB.
If you want to put in a new HD, then flip over the laptop computer and remove the battery. You will see two bays you can open with a philips screw driver. Open them both up, and the one with the box you can easily slip out is the hard drive. Slip the new hard drive, and close it up. Once you restart the computer you will need to install Windows and drivers.