Find or make a neutral grey scale or neutral patch to use as the subject.
A Color-Checker™ has such a scale along one side of the card. The dark tones along with the white patch will give you some idea of the linearity of the source. Sometimes there can be a noticeable colour shift from shadows to peak white. Alternatively, you can choose to use a single pure white patch and varying the exposure time and intensity to compensate in a known way to get a scale of tones if you prefer.
One of the purest neutral white pigments to photograph is magnesium oxide whose spectral response is quite linear over most of the visible spectrum. You can buy tubes of it as a paste from a pharmacy near you and use that as your standard white.
Photograph the "subject" after carefully masking ambient and extraneous illumination that is not from the source you want to test.
Do not use or activate automatic white balance in your camera settings or software.
Make as many photographs as you wish. Keep detailed notes.
You'll evaluate your results comparing them with the neutral standard white patch you made to photograph. Compare only similarly exposed patch lightness to better compare colour shift apart from other factors. You'll want some density so avoid blown-out exposures. Peak white (not highlights) typically falls about 2-1/2 stops over mid grey.
Strive for a neutral environment with no dominant colour to affect your evaluation. To best evaluate your results, set yourself up in an area where the correlated colour temperature of the lighting is 5000K.
ONLY examine your results under continuous source incandescent lighting with the highest colour rendering index you can get (not fluorescent). Quartz halogen will give you a good source if the operating voltage is carefully controlled. Solux makes a 12 volt calibrated 5000K source
The amount of illumination (light level) should be 500 lux at the display plane where you will make your comparisons. At this illuminance and above correlated colour temperature, the eye is most efficient and accurate. Be well rested as your ability to judge colour accurately diminishes with eye fatigue.
The colour of the light can change our perception of colour up to 14 ∆E units.
Just ONE ∆E unit is a perceptible difference in colour!
Always judge colour in the same environment!
Sort your patches according to your colour acceptance. Your notes will help you keep things organized and relevant.
In theatrical lighting, many multi-hued (gelatine filtered) sources are mixed to produce desired effects. In the same manner, you can mix multi-hued LEDs to achieve your ends.
Edit: If you wish to make a statement about the colour rendering index of any specific patch you produce, you'll need a calibrated spectrophotometer as an aid. You can construct one and calibrate it; but, that's another project for another weekend.