Arvind's suggestion is good. You will also need to keep in mind what type of contacts to use to minimize the Schottky barrier. Gold tipped probes and a bit of indium work well.
I do find Arvind's suggestion given above very useful. Microscope will be used to see the contacts.
Its all depend on which thin film you want to study. Majority of the time people use to study the leakage current vs. applied voltage. So, if you want to study thin films having leakage current of the order of microamp or more then you can use simple curve tracer for this study. But if the leakage current is of nanoamp or less then you should use dedicated IV measurement units like Kaithley 42000S etc.
The one of the method you may follow is, have a regulated power supply (like Aplab regulated power supply) for voltage source. Then use any DMM(may be keithley 2000) for current measurement. You may vary the voltage manually and that gives the current.
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