The AD5668 takes a single ground referenced 5v supply, and produces several ground-referenced 0 to 5v output voltages. It interfaces directly to a ground referenced Arduino. If you are going to ride the horse in the direction it's going, you keep those components ground referenced, and supplied with a single 5v.
The AD5668 only produces a 5v swing, and you want a 10v swing. Even if you could shift it below ground, you won't get the swing you want without further components.
The simplest way to build on what you have is to follow the 5668 with an op amp supplied with at least +/- 5v rails (if you use a rail to rail type) or higher than that (if you use an older style opamp).
A gain of 2, with a shift, is quite easy to come by. This is just one solution, which inverts the sense of the output.
simulate this circuit – Schematic created using CircuitLab
Ideally the 5v offset reference and the DAC reference will track, to minimise shifts with temperature or power supply voltage variations, so have a 5v reference which also supplies the DAC reference.
If you still want to use the DAC internal reference (I always do, they're usually good enough for all but the very highest stability), you could dedicate one output channel to providing the reference for the other channels, or indeed use the 2.5v refout feature. Where you use one channel, it would be better to eliminate all the R1, R2 components completely and provide a programmed 1.667v reference directly to all the +ve amplifier inputs. That would give you 7 channels of -5 to +5 control, with just two quad amplifier packages. Where you use the 2.5v refout, divided it down to 1.667v by using R1=10k, R2=20k.