I suggest you to separate and analyse the saponifiable and unsaponifiable fraction of your oils.
Triacylglycerol, fatty acids, waxes, pigments, vitamines, phenols, alcohols and sterols could be useful for a good characterization. Also the volatile fraction (aroma) could be interesting for your purposes.
You can adapt the methods used for analyse olive oils according the Commission Regulation (EEC) No 2568/91 of 11 July 1991 on the characteristics of olive oil and olive-residue oil and on the relevant methods of analysis
Natural oil in the sense triglycerides or volatile oils?. Triglycerides means first it should be converted in to ester form and do fractional distillation. if it is volatile oil means either fractional distillation or column chromatography (if in labscale for research purpose)