How does one construct a cladogram of intraspecies relationships? I have SNP data from several cultivars of rice which I have used to produce alignments, but I don't think that the usual models and algorithms used for generating phylogenetic trees are appropriate, because these cultivars are not the result of speciation events and have been interbred in their histories. How can I best calculate and visualize their degree of relatedness?
Even with inbreeding and other genetic phenomena that might mask actual evolution of these cultivars, any phylogenetic methodology would be capable of determining relationships accurately. 

Try creating a Neighbour Joining tree with MEGA, which is one of the simplest methods available. This should give you enough to check the relationships of the cultivars.