There's nothing actually wrong with your common-emitter amplifier, it's just not suited to driving a low impedance load like a 6ohm speaker. The output impedance is too high (1Kohm) to efficiently drive a 6ohm load. Almost all the power is dissipated in the amplifier instead of in the load - imagine connecting a perfect amplifier through a 1Kohm resistor to your 6ohm load. It forms a resistor divider, almost all of the voltage gets dropped across the 1Kohm resistor and you get a very low voltage across your 6ohm load as you have observed. You'd probably only want to use this amplifier to drive a load that has an impedance of 10Kohm or more.
What you want is to drive your speaker with an amplifier that has an output impedance significantly lower than 6ohms, so you need to add a current amplifier stage - an amplifier which has a voltage gain of 1 but provides a high impedance input which is easy for your common-emitter amplifier stage to drive and a low impedance output to effectively drive your speaker. As above, an emitter-follower amplifier (aka common-collector amplifier) will do this.
Also keep in mind that the output coupling cap will form a highpass filter with the speaker with a corner frequency of f=1/(2*pi*R*C) which gives you 26KHz for your 1uF cap and 6ohm speaker so very little signal in the audible range will pass. You want f to be atleast 300Hz for voice and as low as possible (preferrably around 5-10) for bass/hifi audio. Using a 100uF cap will give you flat response down to 265Hz with your 6ohm speaker.
Better yet, place a Class AB buffer at the output which avoids the need for a large value cap altogether:
The input cap doesn't need to be 100uF. 1uF would do just fine there because its connected to a high(ish) impedance and since f=1/(2*pi*R*C) if you make R larger, C can be smaller.