Okay, so the capacitor is just filtering. That's good because you don't have to match it to anything.
The FFT graph shows the expected results. A square wave is the fundamental plus odd harmonics so 50 Hz, 150 Hz, 250 Hz, etc. makes perfect sense. As the filtering gets better, the harmonics will be attenuated more and the output will look closer to a sine wave.
But now, look at the difference between the output waveform with no load and with your 1K load. It's not a sine wave, but it looks better than the quasi-triangular wave. Yes, it looks a bit square, but the sides aren't straight up and down, they're more sloped. That tells you that the higher frequencies are being filtered out, as expected. It will be more obvious if you change the timebase from 20 ms. to 5 or 10 ms. The problem is that the filtering isn't aggressive enough. That makes sense because you stated that your filter capacitor is only 0.1 uf rather than the 2 uf that's specified. With only 5% of the necessary capacitance you can't expect to see a better waveform. You need to get your hands on the right capacitor or parallel a bunch of capacitors before the waveform will look better. Then it will look closer to a sine wave and the harmonics in the FFT will be attenuated more.
I see that the original article includes a warning that there is no feedback, regulation, protection, etc. in the circuit. Are you sure that this circuit will do what you want?
Ed