Remember this...units can be operated on mathematically just like a variable. So F = k dx is the force of a spring moved dx distance from neutral and having a constant k.
Solve for k = F/dx; now write the units on the RHS of the equation ---> Newtons/meter. This follows because Newtons are the SI force unit and meter is the corresponding length unit.
But what the heck is a Newton? Well, its an arcane term used to describe force, but it certainly doesn't tell us much about the SI units, which are kg for mass, meter for length, and second for time. But, once again, we can figure what a Newton is in SI by setting F = ma; so that the units for force F are kg-m/sec^2 by multiplying the mass unit by the acceleration units. And as a Newton is a force, then a Newton must be one kg-m/sec^2.
OK, then, as the k units = Newton/meter = kg-m/sec^2//m = kg/sec^2 because the length unit (meter) cancels out. And there are your k units in SI units.
The key thing to keep in mind when doing units analysis (which is what we just did here) is that the units can be treated just like variables in an equation.