When to transplant would be in the Fall, after it has gone dormant for the winter.
From what I can see, it may not be easy.  My impression is that it is at least partially surrounded by pavers.  If they are the type that allows water to pass through, it is likely that the root system is partly under the pavers which means you will need to remove some of them.
You will probably need to prune the plant back to reduce the demands on the roots and will most likely need to reduce the roots to get it out of the ground with a root ball smaller the the existing extent of the root system that it now has.  If this is a significant plant to you, you might want to hire a professional to help you with the transplant.  Without knowing the soil that it is growing in and the extent of the roots,  I can not give much more information.  You will also need to know the soil in the location where you plan on placing the plant.  If it is in a well drained area, the new location should be equivalent or it may take some time for it to adapt and grow to its present size.

Depending on the soil type, the roots could still grow under it but not as much as on the soil side.  I am thinking sandy soil which would allow more water to move under it. 
If there is little or no root growth under the concrete, it will be much easier to dig it up.
Do a search for red twigged dogwood root structure on line to get some idea of how the roots grow.  My impression from what I read, is that you will find multiple branching under all that green since the plant will sprout from runners.  Depending on the growth, you might be able to split that plant into several separate shrubs.  Do the search and you will see what I am talking about.

good luck.  That is a nice plant.