The lay person's procedure is: Take your flour and weigh a sample, then wash away the starch and weigh the sample.
Here is a video showing a procedure that you can do at home. I would suggest doing this procedure with a control sample (a flour where you know the protein content to determine your accuracy) before doing with your target sample.
Here is a more scientific approach:
The amount of protein in a food material is usually determined by measuring the nitrogen content of the material and multiplying that value by a factor. The nitrogen content of a given protein varies depending on its source. For milk products a factor of 6.38 is used, for most cereal grains the factor is 6.25, and in wheat products the factor is 5.70. These factors depend on the percentage of nitrogen in the respective proteins.
The flour protein content is an important parameter for bread flour. Flours containing higher protein contents are more expensive than flours of lower protein content. Likewise, flours with very low proteins for cakes are also more expensive. There is usually, but not always, a good correlation between protein content and bakery performance of a flour.
The classic procedure to determine the nitrogen was the Kjeldahl procedure. This involved digesting the sample in concentrated sulfuric acid, then neutralizing the acid with concentrated sodium hydroxide, followed by distillation of the ammonia (derived from the nitrogen in the protein) into a standard acid. The procedure worked well, however it was an environmental nightmare. In addition to the strong acid and base, the catalysts used to speed the digestion included such materials as mercury and selenium. It should surprise no one that the procedure is seldom used today.
The Kjeldahl procedure has been replaced by the Dumas combustion procedure. In the original Dumas procedure the sample is mixed with cupric oxide and heated in a stream of carbon dioxide in a combustion tube packed with cupric oxide and copper metal. The organic material is converted to carbon dioxide, water and nitrogen. The gas stream is led into 50% potassium hydroxide. This absorbs the carbon dioxide and any oxides of sulfur, leaving only nitrogen as a gas. The volume of nitrogen is then determined. Various machines have been developed to carry out the analysis automatically. The percent nitrogen is then converted to protein using the appropriate factor. Both the Dumas combustion and the Kjeldahl procedures estimate the quantity (total amount) of protein and not the protein quality. As discussed elsewhere, the quantity of protein is extremely important in the baking performance of a flour.