You can constrain the group to be monophyletic and then see how much worse the tree is as a result. Most phylogenetic programs should have means by which you can force monophyly on a set group of taxa. You can then examine support for the different topologies using metrics like tree length or Bayes Factors. In a sense, you've already tested for monophyly by doing the analysis on your data, but you can compare the constrained tree to the best tree to see how much worse it is.

You can constrain the group to be monophyletic and then see how much worse the tree is as a result. Most phylogenetic programs should have means by which you can force monophyly on a set group of taxa. You can then examine support for the different topologies using metrics like tree length or Bayes Factors. In a sense, you've already tested for monophyly by doing the analysis on your data, but you can compare the constrained tree to the best tree to see how much worse it is.

Following up Jason's comment, you can do a likelihood ratio test (LTR) from the two hypothesis to test which one is the best one, given your data. However, lack of monophyly can be also an artifact of your data. Check very well the support you have, and be very strict in it.

This is a very open  question! In addition to statistical evaluation of nodes suggested previously (drastically, statistical tests allow you to question the true monophyly of a group when the value is lower than 95%), I would recommend fractioning your dataset and analyzing separately those parts and compare the results. Also, consider adding / removing characters and taxa (data) and comparing the results. These modifications would allow you to determine if the topology you observe is well supported, independently of the number of taxa and of the kind of data, which would be a sign of robustness... Provided that the characters (data) you choose are suitable to the question you address with this analysis. Good Luck!

Last, if you are testing the monophyly of your in-group, keep in mind the importance of the choice of the outgroup, choose at least 3 taxa with at least one that could be considered with resonnable confidence as external to your in-group and run the analysis. Challenge then your in-group or putative monophyletic group with the addition of other taxa in the dataset and see what happen then! Good luck again!

The best evidence IMO are the DNA data. The morphology, esp.among closely related spp, is often subject to parallel or reverse evolution (sometimes embryological data can help discern between more primitive or derived). In human evolution, some characters that are supposed to be derived (e.g. "bipedalism") are more primitive than usu.thought, e.g. the early hominoids might already have been "vertical": humans (esp.for wading-walking?) & hylobatids (for hanging or walking over branches) still(?) have more vertical postures, whereas the great apes might have reduced their vertical spine to different extents (bonobos being more vertical than e.g. orangutans). If a vertical spine is primitive (e.g. Morotopithcus c 18 Ma already?), "bipedalism" does not define a "hominin" (sensu Homo + australopithecines), and human evolution needs a serious revision.

Can you clarify your question?  Specifically, I'd like to know if you want to test the monophyly of your group relative to the other taxa represented in your phylogenetic analysis, or if you want to test for the monophyly of your group, in general.  The latter cannot be tested because it depends on unattainable completeness of taxon sampling.  You might even include samples of all extant taxa in your group of interest, but you can't include samples of all extinct taxa in that group.  If the group is not monophyletic, then excluding any taxon, and its closest relatives outside your group of interest, would fail to detect paraphyly in your group.  Also, as Merladeg mentioned, different parts of the genome, or different phenotypic traits, can have different phylogenetic affinities with other taxa.  In other words, your group might be monophyletic for some parts of the genome, while simultaneously being paraphyletic for other parts of the genome.  In analytical results, this can reflect either analytical ambiguities or true phylogenetic discordance.