My suggestions might sound a little more "boring" but you will be glad you didn't start with flashy rims or leather seats in the long run.
Ive owned a 66, 69, 90, and 65 mustangs (still have the 65). Start with the drivetrain, a car with nice rims or seats won't get you far if the engine doesn't run right, tranny doesn't shift, etc. To get a general idea of the condition of the engine pull a valve cover and see how much sludge is built up on the valve springs etc. After you change the oil, pay close attention to the oil level. You want to know if it is burning ANY oil at all. If it is burning oil then you probably have a problem with a piston ring or valve seal, you will want to fix this first
Start with a basic tune up: oil and filter change, fuel filter change, coolant change, tranny fluid change, rear diff fluid change, new distributer cap and rotor, new spark plug wires, new spark plugs, depending on the carb you might need a new one or a tune on the current carb. Bleed the brakes really well to ensure your safety
Make sure the tranny shifts properly and has fresh fluid. If it is a C4 you might look into a new valve body (easy and cheap).
Thats not everything but you probably get the idea on the tune up
Once you get that stuff out of the way and you have a sound drivetrain, then you can look into some mods to have fun with. Personally I would start with front disk brakes if this car currently has 4 wheel drums. 4 wheel drum brakes are scary if you need to stop in a hurry, especially if it is raining. There are lots of front disk conversions out there for a 65. You can get brakes from a ford granada which work great or you could look into kits from baer, wilwood, etc which will run you quite a bit more $ than granada brakes. Again, no matter what you do with the brakes, bleed them really well to make sure that there is no air in the system. If you have drums at all, consider new wheel cylinders if the current ones are leaking at all. You can also add a power brake booster and dual master cylinder to help braking. Tires also play a huge part in brakeing, make sure that there is a good amount of tread left on your tires and that they are properly balanced
At this point you will have a healthy drivetrain and safe brakes and you can have a little more fun with mods. The typical place to start would be with the exhaust to get her to exhale better. Start with headers of your choice and 2.25" or 2.5" dual pipes and mufflers of your choice. To get her to "inhale" better look into a 4 bbl carb if it doesn't have one, 14"X3" air filter and an aluminum intake.
After these mods the possibilities are endless, cam and lifters, aluminum heads, forced induction, etc.
Have fun with your 65 and unless you have a K code, mod it however you want, don't worry about how it came from the factory in 1965
Oh lastly, don't replace the struts as was previously posted. The main reason for this.......a 1965 mustang does not have struts