Understanding dependent/independent variables in physics

How does one determine the independent and dependent variables?

It's totally relative. In $F = kDelta l$ all three variables can be considered dependent or independent, depending on your purpose.

E.g. in $Delta l=frac{F}{k}$, $Delta l$ would now be considered the dependent variable.

Now suppose you studied a set of different springs, so that:

$$k=frac{F}{Delta l}$$

$k$ is 'normally' the proportionality constant (or factor) but in that study it would be the dependent variable.

In an equation there is no inherent distinction between dependent and independent variables. There are to my knowledge only two contexts where the distinction makes sense.

Experimental: In an experimental context the independent variable is the one that the experimenter is controlling in the experiment. It is the treatment. For example, if the experimenter is determining the resistance of a resistor using a series of a few different voltages, then the independent variable would be the voltage and the dependent variable would be the current.

Statistical: In a statistical context the independent variable is the one that is known with no error. That is usually an approximation, so instead the independent variable is the one with negligible error. If none of the variables of interest have negligible error then unusual statistical methods must be used.