Not quite.
Before (preferably) or after you have bought your aeroplane, you will need to find an established operator of your type, willing to add it on his AOC.
This is usually done at a cost but perhaps if you have something of interest to them, you can trade the use of you plane Vs a financial arrangement.
Once that's done, since you will be under the AOC holder's responsability, it will be down to them to decide whether or not as a pilot you can operate public transport flights.
Their AOC might require a minimum number of hours, experience on type or whatever else and therefore dictating your ability to fly your own plane as PIC.
Of course if you have a plane, customers and the minimum experience required (which includes 700 hours mini for single pilot public transport according to the JAR), the rest should be a breeze.
Like I said, buying an aircraft is easy but it's the rest that will cost you time, money and some frustration.
I think you will find thatDaiflycan come up with better info than mine as his knowledge is quite vast when it comes to the business side of A/C operations.