I've found that solubility studies can be very dependent on the analyte crystal size, and good, consistent mixing. If you are not certain, choose a cheap compound with similar properties, and practice with that analyte first to ensure that you get the same values as published literature.
I prefer to use ultrasonication and vortex mixing, with some warmth, then allowing the solutions to cool. I also reanalyse aliquots of the solutions several times, to ensure each result is consistent. If I see variation, I know that the dissolution or clarification methods are not working properly.
To clarify the solution, I prefer to use strong/quick centrifugation in a sealed tube in a controlled-temperature centrifuge, without any filtration.
Longe centrifuagtion times in some centrifuges may change the temperature of the solution.
If you filter, ensure that only pressure filtration is used. If you use vacuum filtration, the solvent can evaporate, cool the solution, and more analyte precipitates.
Provided you have crystals present at all stages, as recommended by Marina1111, you are extremely unlikely to form supersaturated solutions.
If you want to be certain, you can sprinkle a few crystals onto the solution, and gently swirl as they descend. Then centrifuge, sample the supernatant, and dilute.
Like Danko, I would not use chromatography for solubility testing of a pure compound, but some other property - eg spectroscopy, optical rotation etc. If you do want to use chromatography, you should dilute samples and standards with a solvent that will completely dissolve the highest concentration of the analyte.
Please keep having fun,
Bruce Hamilton