the easiest way would be to fly from one side to the other, the cheapest way would be to walk, although you'd have to factor in the accommodation and food required for this longer journey which may bring the total to more than the cost of the plane ticket.
Wow, that first answer must have really helped ;-) Bunac has a website which might help you with visa etc if you're young/ a student: http://www.bunac.org/uk/canada.asp
If you can find someone to offer you a job before you leave: http://www.cic.gc.ca/english/work/index.html
There won't be much pick-up work for illegals in Canada, I don't think, as the economy isn't great enough to warrant a thriving job market, but you might strike lucky. Beautiful country.