Great job. The main assays to conduct can be:Gas  omposition in the package overtime,physicochemical, nutritional and microbiological quality parameters of the fresh produce and more importantly the sensory parameters of the same peoduces over storage time. Temperature management remains to be very important.

It depends on whether you want to measure the packaging material, in which case you measure permeability to gases (oxygen, carbon dioxide, perhaps ethylene) and water vapor.  Or you may want to measure the quality indicators of the produce over storage time, which can be physical, chemical and microbiological factors, which is dependent on the produce.   

Since you are assessing the effectiveness of a certain packaging material, several fresh produce should be checked. As it might differ according to the produce. When considering quality, Fresh weight loss is an important indicator. Colour change, firmness, visual quality or certain chemical compound would be ideal, including gas permeability measurements as suggested above. And finally, it really depends on the produce. What I have suggested can be used mostly for fruits and vegetables or any other relevant material. 

Packaging integrity test could be added along with gas composition as well as other parameters above. Depend on your product pH and cook loss can be added, for meat for example.

What I learn from your question is that you are looking for information about packaging materials and not necessarilly packaging conditions. So I think and based on the produce that you want to protect, a suitable polymer should be selected. This selection, itself depends on mechanicl/barrier properties of the polymer. For example for packaging of mushroom, you need a combination of PS and Saran (PVDC), or for short storing of red meat you need foamy PS at the bottom along with HDPE as cover . The situation can be more complicated when you are going to use MAP. So I think while looking at the characteristics of the produce (i.e. respiration rate, microbial load, light sensitivity, etc.) you have to choose your material based on its own characteristics. Some polymers are resistant against moisture and not gas, whereas some are able to stop migration of the both. If you like please let me know what produce you are going to put in a package then I can be more specific. 

Water vapour permeability test is essential for fresh produce packaging material which decides its transpiration and firmness. Also burst strength, puncture test to be carried out for packaging material for its durability.