Can I upgrade the Motherboard on my Optiplex GX620 to support a core processor?

You cant use ANY core processor.  You can use a Pentium D 915 after upgrading the bios to A11.  The Pressler 900 series works up to 960 in the tower model.  Don't be fooled by the Mhz Myth.  A Pentium D will be much improved over the original Single Core Pentium 4.

See the FAQ

GX620 FAQ

I bought the refurbished PC upgraded to a Pentium (R) D 3.4GHZ with 4GB of RAM and a 64 bit operating system, but the company I work for online says it is benchmarking to low for me to use with them.  Is this as high as I can upgrade this processor?  Can I overclock it perhaps and if so how?  They say it is benchmarking at 750 and they need it at 1500. I have no idea what these numbers mean though.

Mhz isnt going to get anything.  Overclocking is not an option with ANY optiplex model.

You wont get double performance from the cpu EVER even at 4.0Ghz.

The GX620 will not take 4th Gen Core I7 processors.

I assume the numbers they are tanking about are the points on passmark benchmark, you can find the list of every processor associated with it's passmark result HERE

But as SpeedStep says you can't upgrade the cpu on your motherboard over the Pentium D 960, that is still not enough for your needs. So the only thing you can do is buy a new pc or a new case+mobo+cpu bundle. Overclocking can't give you double performance, so you can forget about it. 

The 1500+ NUMBER if its passmark has Zero to do with Mhz and everything to do with architecture.

http://www.walmart.com/ip/Refurbished-Dell-OptiPlex-755

Meaning even a 1.8 Ghz Core 2 Duo Beats a 3.4 Ghz Penitum D by almost 100 percent aka its 2x faster.  Netburst 775 processors dont break 1000 even at the D 960.  3.73 Ghz Quad Core pentium D is 985.  Lowest that I would Recommend is Optiplex 755 with Core 2 Quad Q6600.

3.0 Ghz Pentium 4 is at the bottom at 358. Whereas Core I7 @ 1.60 Ghz is at 3052.