Ok, gear head, I am considering to sell a kidney to buy a 2005 Mustang.
Any ideas how to beef it up?
Lots.
What engine is it coming with? If it's not the V8, then the first step in beefing it up is to yank the V6 and swap in a V8. That's sort of a pain, so I'd strongly recommend starting with the V8.
After that, engine power amounts to how much air/fuel you can get into the cylinders and how efficiently you can pump the spent gases out.
The usual starting point is exhaust. Get rid of the factory exhaust and put on a cat-back exhaust. This replaces everything behind the catalytic converter. Do not remove the cat. It's illegal. And stupid. You can absolutely make more power than you can ever use without turning your car into a smog bomb. My old '99 Corvette makes 815HP/705TQ and will pass any sniff test in the country.
After the cat back, the next step is to replace the exhaust manifolds with tubular headers. They're more efficient at scavenging gases. If you use shorties or mid-length headers, you'll stay smog legal. And unless you're building a full-on pure race engine, they will be more than adequate.
Then switch over to the intake side. The '05 Mustang will benefit from a cold air intake to allow more air in, and since it pulls in cold air (the factory intake is in a place that pulls hot air...) it will be denser.
That's all relatively cheap and easy. After that, things get a little more complicated. You can pull the heads off and have them ported and larger valves installed. You can swap in a more aggressive cam. If you want oodles of power, add boost or nitrous. Nitrous is relatively cheap, but the size of the tank limits how much you can use it. Blowers and turbos (I prefer blowers for street cars) cost a lot more, but they're always available when you need (ok, want...) the power.
When you increase the go, you also need to think about the whoa. Upgrading the brakes is a good idea. You can start small (as you do with engine mods...) and change to ceramic pads. They're noisy, but stop faster. If you add a lot of power, then look at big brake kits from Wilwood or Brembo. My Vette has Brembos.
At some point, traction will become a huge issue. That's when you need to swap wheels & tires. I don't know how much wider you can go on that model, but a little google-fu would tell you. And any wheel manufacturer will have that info at their fingertips. Tires are a compromise. If you don't need to drive in bad weather, go with a full on street legal race tire. I like the Nitto NT05 (which is on my Vette) or the Michelin Cup 2. I drive the Vette in the rain, and with reasonable caution the Nittos do OK. I do not drive in the snow. Ever. Not even close. It would be like driving an 800HP tobaggon. That, and the blower is built to such tight tolerances that driving in freezing weather has the potential to contract the housing enough to contact the impeller. And then it's a $5000 boat anchor.
I also recommend that any car with significant mods be equipped with a roll bar and 5- or 6-point harnesses. Race harnesses are not technically street legal anywhere in the US, but they are undeniably safer, and although I have been stopped once (OK, maybe twice...) for bending the speed limit, no officer has ever batted an eye at mine. Maybe one of the LEOs on here can chime in on that subject.
There's also plenty of work that can be done on the suspension. Swapping in a set of coilovers can improve cornering and will let you adjust the ride height. A strut tower brace and sub-frame connectors stiffen the car, helping it stick to the corners better, as well as preventing flex under hard acceleration. Yes, you can twist the body, with enough power and traction. After a while, the doors won't open or close properly.
