What can be the cause of my car vibrating at high speed between 60-70mph?

Hello, it sounds like you have two concerns: first, vibration cruising or accelerating while driving, then vibration braking. First: the cruising/driving. High speed vibrations are typically caused by out-of-balance wheels. A simple high speed spin balance can be performed by shop with tire installation equipment to repair this. At the same time, if there is something inside the tire like "Fix-a-flat" or similar, this must be removed and cleared out of the tire, because that liquid will create an out of balance situation. A second possible cause of the cruising vibration could be related to axles that may be out of balance. Start by checking the wheel balance first, though, and progress from there. Second, the braking vibration is most likely caused by out-of-round brake disc rotors. This could be due to multiple causes, but all of which indicate that the brake temperature has been higher than the rotors can sustain without warping. Ensure you are changing your brake fluid every 3 years -- fluid that is contaminated (as naturally happens) by moisture will overheat quicker, and allow your brakes to be too hot too. This can cause the rotors to warp. Also, brake pads that are installed on rotors that are not properly machined can cause a vibration to occur. Have your braking system inspected thoroughly to troubleshoot this.