Colchicine can be applied as a cream to the growing part of the plant. Or additionally, the seeds may be soaked in the solution before sowing them. These are crude methods and what may be used in non-laboratory conditions. Truly speaking, to induce polyploidy, one needs to 1st culture the cells on culture media. these cells may be plated on a media which contains colchicine and then sub-culutred onto a regular growing medium after allowing the cells to divide once or so. Each stage must be monitored with microscopy.
Alternatively, somatic cells of the parent plants may be fused by a process called SOMATIC CELL HYBRIDIZATION to induce polyploidy. This is much more complex a procedure.
As far as Colchicine goes, as pointed out by BKirk783, it acts as a microtubular inhibitor. It is actually an alkaloid derived from plants. It can be used in treatment of cancer as the cancer cells are more active due to rapid multiplication - it thus stops the multiplicating cells and induces a static change in the cancer cells allowing them to be phagocytosed or eaten away by the body's immune system. However, due to availability of better, less toxic drugs, it isn't used for the purpose as higher doses are required. It is however used in gout in humans in lower doses. Other use includes cytogenetics which is what we've been discussing so far. Besides colchicine, another alkaloid called vincristine and vinblastine, both Vinca alkaloids are also used in cytogenetics.