Dell, as usual, provides less than helpful documentation. Its Comprehensive Specifications document only states that your PC supports Celeron, Pentium Dual Core, Core 2 Duo, and Core 2 Quad. http://www.dell.com/support/home/us/en/19/product-support/product/inspiron-560/manuals
Ask this question in Dell Forums. Someone will have a list of supported CPUs or at the very least, they can tell you which quad-core CPU they have. http://en.community.dell.com/
Your biggest bang for the buck would be to replace the hard drive with an SSD, but that would involve reinstalling the operating system from the original discs. I'm not a fan of cloning anything other than a fresh copy of Windows.
According to the product specifications for your Dell Inspiron 560, integrates an Intel(R) G43 chipset.
Please try the link below for processor compatibility information:
Please bear in mind that processor compatibility depends on the computer manufacturer, I would recommend contacting Dell for more information.
I have no detail information about the mobo within your Inspiron 560, but after searching it on Dell website, it might use Intel G43 chipsets, in other words, all the Core 2 Quad processors would be supported by this chipsets. But there is another more important problem, the maximum CPU TDP supported by your mobo. Most Intel Core 2 Quad processors' TDP exceed 95W, if your mobo just support up to 65W, even if that processor would work but there is the hidden over burning risk, which might cause both of your processor and mobo out of work permanently. So I suggest you to find Intel Core 2 Duo with TDP of 65W, such as Q8200S, Q8400S, Q9400S, Q9505S, Q9550S, but those processors are very rare not easy to find. But most mobo with G43 chipsets support processors with TDP up to 95W, if yours is so, you could have a larger range to choose, such as Q9650, the best.
I wish you good luck!