Determining the correct amount of bulk decoupling capacitance for a PWM LED driver
VA rating on 12V supply must exceed stripled Watt rating.  These are design for automotive 14.2 at max brightness.  These are not highly efficiently with a threshold of around 9V and are rated in Watts per foot or per meter.

The problem with many wall wort supplies is the internal cap may not be rated for the pulsed ripple current you intend to apply and prefer to be run at constant load .  

You can use a surplus PC PSU at 12V or perhaps a universal laptop charger which is less robust but may have a selection for 15V at 4A or more.

to choose a cap for unknown flakey supply, it must handle the RMS current you intend to switch. So if 5App 50% assume 2.5A and choose a cap rated >3A rms ripple current.

to choose c value at 1K or 10kHz compute C for 10 cycles using R=Watts/(I^2) as the load is quasi linear with series R's.

e.g at 1kHz  and RC=10cyc/1kHz=10ms or 1ms @10kHz


thus if strip leds are 5A, for cap sag of 10%V in 1ms @10kHz, 
Ic=Cdv/dt so C=5A*1ms/1.4V=3600uF rated at >=16V


3600uF/16V is not avail, but 2200uF is avail here. Ripple current =8.1Arms radial leaded $1.5 , get two


1kHz would need 10x bigger cap.
Mosfet should have RdsOn&lt;&lt;100mOhm@5A for no heatsink @0.5W dissipation, lower is better and can be driven by NE555 PWM clock.