There are many ways to determine whether a polymer is hydrophillic or hydrophobic. Firstly, you can determine if a polymer is inherently hydrophillic or hydrophobic just by looking at the chemical structure of the polymer repeat unit. Take polypropylene for example, which has CH and CH2 groups along its backbone and a CH3 pendant group. It is a very non polar molecule which means that inherently, it will be hydrophobic. In the chemical world, like molecules are attracted to each other, so polypropylene is an oil loving material (oleophilic) because it's repeat unit has a similar structure to oils (hydrocarbons).

On the other hand, take Nylon or cotton, these polymers are polar because they contain a carbonyl group and hydroxyl group, resepctively. Therefore, inherrently Nylon and Cotton are hydrophillic. Nylon has a moisture regain of 4%, cotton is even higher.

Now in the material world we can control these properties by modifying the surface properties of the polymer. So an inherrently hydrophillic polymer can be modified to have hydrophobic surface properties and vice versa. This can be achieved by changing the surface roughness of the polymer or applying functional coatings.

The simplest experiment to determine the surface properties of a polymer is the water contact angle measurement. It is important to have an understanding of the surface properties of your material so that you can choose the most appropriate model for your results (Wenzel/Cassie Baxter etc). So you could perform AFM or profilometry before any water contact angle measurements. Finally, it is important to understand that there can be significant variability in the data from a water contact angle measurement so you need to determine whether to measure static contact angle or dynamic contact angle.

Alternative tests include wilhelmy's method and the parachor method which estimates surface tension from group and structural contributions.

Prepare thick films on substrate and do contact angle measurements by water

You need to use BET instrument which will estimate the surface tension of the polymer through calculating its contact angle, the higher the contact angle, the higher is the hydrophilic nature of your polymer sample, and the opposite shows its tendency to be hydrophobic.

There are many ways to determine whether a polymer is hydrophillic or hydrophobic. Firstly, you can determine if a polymer is inherently hydrophillic or hydrophobic just by looking at the chemical structure of the polymer repeat unit. Take polypropylene for example, which has CH and CH2 groups along its backbone and a CH3 pendant group. It is a very non polar molecule which means that inherently, it will be hydrophobic. In the chemical world, like molecules are attracted to each other, so polypropylene is an oil loving material (oleophilic) because it's repeat unit has a similar structure to oils (hydrocarbons).

On the other hand, take Nylon or cotton, these polymers are polar because they contain a carbonyl group and hydroxyl group, resepctively. Therefore, inherrently Nylon and Cotton are hydrophillic. Nylon has a moisture regain of 4%, cotton is even higher.

Now in the material world we can control these properties by modifying the surface properties of the polymer. So an inherrently hydrophillic polymer can be modified to have hydrophobic surface properties and vice versa. This can be achieved by changing the surface roughness of the polymer or applying functional coatings.

The simplest experiment to determine the surface properties of a polymer is the water contact angle measurement. It is important to have an understanding of the surface properties of your material so that you can choose the most appropriate model for your results (Wenzel/Cassie Baxter etc). So you could perform AFM or profilometry before any water contact angle measurements. Finally, it is important to understand that there can be significant variability in the data from a water contact angle measurement so you need to determine whether to measure static contact angle or dynamic contact angle.

Alternative tests include wilhelmy's method and the parachor method which estimates surface tension from group and structural contributions.

A little modification: as per my previous answer the words hydrophilic and hydrophobic positions in the sentence should be ex hanged as higher contact angle is related to hydrophobicity.  Moreover, the contact angle can also be determined using a contact angle goniometer.

All the previous answers are most acceptable. Still there is limitation conclude with contact angle measurement. Because polymer film morphology also effects the contact angle value because PVDF even it is hydrophobic  in nature shows low contact angle value which depends on the parameters of film making like as thickness, concentration, solvent used for film making. Means if the film is more microphorous nature it shows low contact angle, we get more information from membrane technology papers.

Solubility is one of the better option for reveling of the hydrophlicity/ hydrophobicity.

Most of the hydrobobic polymers are soluble in non polar solvents and polar solvents. Whereas hydrophilic polymers are soluble in polar solvents only.

While as previous persons explanation chemical structure also give more information...

Look for the functional groups present, for example, -COOH, -NH2, -OH etc. The more you have, more hydrogen bonding between water, hence the polymer will be hydrophilic and water soluble. 

If you have more hydrocarbon chains (-CH2) and aromatic units (benzene etc) without those common functional groups, then the polymer will be hydrophobic and water insoluble. 

synthetic polymers can be made hydrophobic or hydrophilic based on the type of monomers you choose for polymerization.

Contact angle measurement is very basic and easy to perform with polymer thin films. In case of bulk samples, simple dissolution in water will give indication about solubility. Depending on the nature of the polymer, either it will dissolve completely or will swell to for a gel or will not dissolve..

Irrespective of functional groups polymer has hydrocarbon chain as well. So can we say that all polymers ( hydrophilic and hydrophobic ) can interact with organic molecules ?

Hydrophobic polymers are non-polar in nature.

Roughness parameter are also important in assessment wettability of the materials. Maybe this article would be helpfull: Kubiak, Krzysztof, Wilson, M.C.T., Mathia, T.G. and Carval, Ph. (2011) Wettability versus  roughness of engineering surfaces. Wear, 271 (3­4). 523 ­ 528. ISSN 0043­1648 

Hydrophilic means water loving; hydrophobic means resistant to water.

Hydrophilic polymer get absorbed or dissolved in water, while hydrophobic polymer only dissolve in oil-based substances.

Hydrophilic molecules are polar and ionic; hydrophobic molecules are non-polar.