Hi!
It looks like your spine is following your spine controllers, but you spine controllers arent following anything at the moment.
Basically what constraints do is overwrite all the hierarchy above the child object when it comes to the channels they affect, which means that your joints will stand still in 3D space if your Controllers stand still in 3D space. This is exactly what you want. But now you just have to figure out the hierarchy to make those controllers follow their own hierarchy.
For that you have two main approaches:
1) hierarchical, you basically just parent the the controlls to the controls higher in the hierarchy.

This has the upside that it computes really fast and is easy to make, but it has the downside that it makes parent switching more difficult and you'll have to put IK CTRLS really high into the hierarchy. Also you'll have to decide wether you'll use offset groups or freeze transforms to zero out your CTRLS.
2) Constraint/Modular, which means that you give your modules a basic hierarchy and then establish their relationships with constraints , which allows for parent switching. You'll have to introduce constraint groups though, group nodes above the controller that are at the same space in 3D which get constrained. Because you don't want constraints on your CTRLS directly.

(Really overcomplicated example sry, but that was just the Rig I just had open.)
Most riggers use a combination of both approaches.
If you want to know more about correctly structuring rigs, I'd really recommend this series on the Maya learning channel.
https://www.youtube.com/watch?v=MV4XRgmTynY&list=PL8hZ6hQCGHMXKqaX9Og4Ow52jsU_Y5veH
It's a bit older but it still mostly works like that, and it is really well explained.
I hope this helps!