Does the electrical gradient always oppose the movement of charge?
It doesn’t always oppose it. It depends on the sign of the charge.
I’m assuming here that by electrical gradient you mean the direction of the electrical field,
The convention is that field lines are drawn in the direction in which a positive particle experiences a positive force (for electrical and gravitational fields anyway) so the field direction is in the direction that positive charges tend to accelerate. Thus it opposes the direction in which negative charges will accelerate.
In summary, the field could be said to be “downhill” (following the arrows on the field lines as they are drawn) for a +ve particle and “uphill” for a -ve particle.