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?
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. 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.
While the UK is a Visa Waiver country, the VWP status has nearly zero advantage for crossing into the US by land, because of the need to park your car, get out, line up, and then fill out an I-94 form (which you also need to pay for). On a busy day, that can easily add an hour or two to your border wait time (especially during an era of currency parity when every Tom, Dick, and Harry is making a beeline for cheaper shopping).
Canadian passport holders only need to show their passports at the booth and then they're on their way - no need to get out of the car.
Get your Canadian passport - it's well worth the fee. Also, I believe Canadian citizens are required to use their Canadian passports to enter and leave Canada, even if they hold other citizenships. The law isn't really enforced, but better safe than sorry.