Quick question : you have an e3-1225 V5 , right ?

https://ark.intel.com/content/www/it/it/ark/products/series/88210/intel-xeon-processor-e3-v5-family.html

There are no hex core e3 v5 processors. What you could do is move from a 4 cores 4 threads ( the one you have now ) to a 4 cores 8 threads like the E3-1245 v5 , it's around 85/90 usd on ebay.com

It's definitely a cheap upgrade and a fraction vs buying a newer (or current) unit. 
Now the question is if your workloads take advantage of the extra 4 virtual cores vs you needing 6 real cores instead, and that's something only you can know

How much would it cost if I would replace my E3 processor with an E5 processor including the motherboard?

@Amruth 

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You cannot change to V5 versions.

The only upgrades are QUAD core max.

Intel® Xeon® Processor E3 v3 Family

Dell case etc are custom, is not a situation where you can swap in a different motherboard so easily.

You would need a different model mb , prob a different case , a different psu 

The Intel Xeon E3 V3 processors are the Xeon counterparts to the Core i7/i5/i3-4000 Series 4th Generation Core processors. The Dell Precision T3620 workstation uses Intel Xeon E3-1200 v5 and v6 processors and the Core i7/i5/i3-6000/7000 series (6th and 7th generation) processors. A BIOS update to the latest BIOS is required to use the latest versions of these processors. My T3620 is running a 3.6 GHz Core i7-7700K processor with no issues and am planning to upgrade to a Xeon E3-1275 v6 processor and 16 GB of ECC memory as soon as I have the money for it.