First of all my advice is you must refrain from trying out a Poisson distribution just as it is to the data. I suggest you must first make a theory as to why should Poisson distribution fit a particular dataset or a phenomenon.
Once you have established this, the next question is whether the distribution is homogeneous or not. This means whether all parts of the data are handled by the same poisson distribution or is there a variation in this based on some aspect like time or space. Once you have convinced of these aspects, try the following three tests:
- likelihood ratio test using a chi square variable
- use of conditional chi-square statistic; also called poisson dispersion test or variance test
- use of the neyman-scott statistic, that is based on a variance stabilizing transformation of the poisson variable
search for these and you will find them easily on the net.
