Frequency Synthesizer via Harmonic Filtering?

I don't know if this would work, but...

Let's say I make a bandpass filter with a high Q and a pass frequency of 5 kHz. If a use a 1 kHz square wave at the input, will I get a 5 kHz sine wave at the output? This is based on the idea that a square wave can be represented many harmonic sine waves.

Is there a better/easier way to multiply frequencies?

That would work. Square waves have strong odd harmonics, so 5Khz will be easy to filter out.
Better/easier depends on your application. ie: You could just generate your 5Khz directly with a sinewave oscillator. 5Ghz is a different story.

You'll probably need to amplify the output signal after filtering, square waves contain a lot of harmonics but for ever harmonic you go away from center the power decreases pretty fast. Mind you you'll also only get odd harmonics. There are better ways to synthesize frequencies, especially with micro controllers and modern DDS chips. Even a realtivly cheap micro controller should be able to output almost any audio range frequency, maybe up to a hundred kilohertz or so with pretty fine frequency control.