I am a Canadian citizen with a British Passport and I live in Canada and I am planning on traveling by car to Buffalo USA and then boarding a plane to the West coast for a holiday. Will I have any problems crossing the border?
Given that you are a dual citizen (British and Canadian) you can get a Canadian Passport at one of the Passport Canada Offices, which there are many around Toronto in 24 hours for C$110 and then as a Canadian citizen travel to the United States with no issues of Visas, ESTAs, VWPs or similar.
No, you won't.
Or at least, not more than any other British citizen crossing the border, which is to say you won't be denied entry but it will take you longer than a Canadian passport holder.
At the border you will be treated as a British Citizen, which means fingerprinting, a greater degree of scrutiny and some forms to fill in, and a payment. Remember to take some US currency with you to pay the visa waiver fee. With a Canadian passport you would probably be waved through after some straightforward questions. I've never tried presenting my Canadian citizenship card as well as a British passport, but you might try that.
Not an answer, but I strongly recommend getting a Canadian passport. It stops you having to put up with these inconveniences, and allows you to apply for a NEXUS or similar card which will speed you through the whole process.