Simple Voltage Inverter without using IC
Your solution is quite close. However you can take advantage of have 2 low ohmic output phases of your oscillator and build a charge pump. The schematic would be as the following,







simulate this circuit &ndash; Schematic created using CircuitLab

Where PhiA and PhiB connected to each phase of your relaxation oscillator.

On the charge phase of either pumping cap you loose 1 diode drop, 0.7 V. On the discharge phase you loose 1 Vcesat plus 1 diode drop of 0.7V.

So for a 1 stage charge pump as you have shown, you will get under no load approximately -(5-0.7-0.7-0.2) ~ -3.4 V. 

For the two stage charge pump its approximately double that at -6.8 V.