I'm not quite exactly sure what you're asking, as there are several potential levels of complexity in interpreting this question.

If you're trying to identify a pure element, then it would seem like you'd want to measure as many physical properties of the material as you could. For example, depending on its phase, you might measure its melting, boiling, or sublimation point. You could also measure its density and heat capacity. Then, after identifying some physical characteristics, you could match up these properties to known properties of elements to find the atomic weight.

If the element can easily be turned into a gas, then you have an easier route available. You can turn the material into the gas phase, and then use the ideal gas law (PV = nRT). First, you would determine the mass of the unknown element. Then, after turning it into a gas, you would measure the pressure, volume and temperature of the resulting gas. R is the gas constant, so all you would have left is the number of moles, n, which could then be calculated. Once you have the number of moles, you could then calculate the atomic weight by:

atomic weight = (starting mass) / (number of moles)

Of course, there are also other techniques that could be used. Historically, for example, the atomic weight of elements was initially studied by determining in what proportions things react with other elements. Also, experimental setups using electrolysis could also be used for some types of metals. Modern techniques can utilize x-ray diffraction if the material is crystalline, or types of mass spectroscopy. The exact methodology you would probably use depends on what information you have available to you, as well as what kind of element you're examining.