How do I determine the appropriate likelihood function for a custom CDF I'm doing some work describing how things dissolve in solution, and I've determined that a particular parameterization of the 3P Weibull provides a good fit. Now what I want to do is specify target amounts dissolved at a particular time and use MLE to find the values for the parameters. Since there isn't just one 3-parameter Weibull distribution, I can't just turn to R or Python to get their pre-defined distributions. Instead, I need to figure out how to determine the likelihood function (negative log likelihood?) that I should pass into a minimizing function.
The CDF I'm using is $a * (1-e^{-(x/b) ^ c})$ where a, b, and c are the parameters and x represents an amount of elapsed time.
Is there any direction on how I go from this CDF (I don't know the PDF) to a likelihood function? Do I even need to assume this is a distributional problem? Is there a better way to do what I'm trying to do?

Before you get to the likelihood function, the first issue here is to confirm that this is a valid cumulative distribution function.  If you examine the requirements for this, you will see that you must have $a=1$ for validity of the form, so your distribution will reduce to the two-parameter Weibull distribution.
Once you have narrowed down the CDF, you then obtain the PDF using differentiation and use the latter to form the likelihood function for an arbitrary sample.
