The bus trip is very slow and uncomfortable. I would advise you to take a train from London Kings Cross Station to Edinburgh. They operate frequently and the journey takes about 4 hours 20 minutes.
The cheapest tickets are those booked well in advance for specifically timed trains. Booking opens 11 weeks ahead of your journey
www.eastcoast.co.uk or the ticket office at Kings Cross Station
A railcard will save you a further 34% www.railcard.co.uk
Edinburgh is 400 miles away, bus (coach) will take at least 9 hours
http://www.nationalexpress.com/home.aspx
Or take a train or aeroplane at greater cost but much quicker
Check airline prices
QUicker by train