Yes you can - simple hand tools - cut off old plug - strip back outer insulation and fit new plug paying proper attention to where the wires go and the correct fuse if it's a fused plug - 13A in UK.
More worrying though is why the original one melted - it obviously got too hot - it may have been badly made or perhaps the socket it plugs into is somehow defective.
Have someone confident with plug wiring fit the new plug for you - try to discover why the original one melted - do not leave home with the dryer running until you're sure it's safe to do so.
Good luck... Take care...
Peccavi
February 2010