Inspect the frame of the car. Make sure it is not rusted. Make sure it is not bent. If you don't know how to do this, then have someone who does do it for you.
Cars nowadays are unibodies, built out of giant three-dimensional body parts that fit together kind of like Legos. Back then, though, cars were built on frames, a large, rectangular piece of solid steel. Everything is adhered to the frame -- the body, the engine, the suspension, everything.
If a frame rusted out, that was the end of the car. When the frame rusts through, the car loses its structural integrity, even becomes illegal to drive. The car can literally break in half under its own weight.
If a frame got bent, usually from an accident, that was also the end of the car. If you are in an accident and your frame gets bent, your car is totaled. There's no fixing that. Sure, there are places that specialize in frame straightening, but it seldom works and never works fully. When the frame is bent, the car will not drive down the road straight, like if you are driving in a straight line through snow, you will see four sets of tracks, not just two, because your rear wheels aren't directly behind your front wheels but creating a second set of tracks to the right or left of the tracks left by your front wheels, meaning your car is going down the road crooked. That's so dangerous that it makes that car illegal to drive.
Since you're not saying much about how much you're going to spend, I can't give much more advice than this. But this advice is key. Everything else on that 65 Mustang you can replace or fix. Not the frame. If the frame is bad, the car is junk. There's no fixing it. So have the frame inspected.
You asked about how many miles is too much? There's no way to know. Cars back then only had odometers that went to 100,000 miles. When they hit 99,999.9 miles, the odometer returned back to 0 miles. A car that old, you can't possibly know how many miles it has. Maybe the odometer says 36,423 miles, and the guy selling it is swearing up and down that that's all the miles it has, but you can't know that. It could have 136,423 miles, or 236,423, or even 836,423 miles, or the sky's the limit. A car that old, you can't look at its condition to guess what its miles, like you can't say, "Well, it doesn't look too banged up, so 36,423 miles is probably right or maybe 136,423, but not more than that. You can't know that. So forget about mileage. You can't use mileage as a guide when buying a car that old.