Electronic – MC34063 issue. Switches off every cycle

The circuit is fun to follow but very very very unconventional.
If you submit circuits like that Olin will have a triple fit and explode on the spot :-). The zig zagging layout style and having current flow and control "flow" in both directions makes the diagram very hard to follow, and the bits that you have left out also do not help.

BUT

Most of it seems possibly OK.

• The Pololu buck converter information here accepts a wide range of inputs voltages and has an adjustable output voltage of 4 to 25V at 300 mA max.

• The drive to M1 is ingenious but not certain to give good results. M1 should have a gate signal of no greater than 12V when it is turned off and a gate signal of between about ground and 6V when turned on. Your arrangement achieves something like this but relies on the supply rising to 37V when the LEDs are off (so 1/3 of 37 ~= 12V. and when LEDs are on and Vload supply ~~~= 24 V Vgate = 24/3 = 8V. This gives 4 Volts of gate drive which will work for some FETs BUT it is based on an ill defined loading of the power supply.

• The FET that you have chosen datasheet here is a good one and the gate threshold voltage about matches your control method. It will probably but not certainly work in practice.

• A more reliable method would be to use the PIC pin 7 to drive a small "jellybean" npn transistor - say 10K from PIC pin 7to NPN base. NPN emitter to ground. Then connect NPN collector via a say 1K to P Channel FET gate and connect a say 10k to 100k resistor between P Channel FET gate and source.
  Supply FET source from 12V from buck converter as already planned. The FET is now driven by the PWM signal and you will also get dimming of your 1 Watt 12V LD strip if PWM is applied to the original LEDs.

• If you do not want dimming you can add a say 1 uF or so to the NPN base and a diode across the base drive resistor. The capacitor will be charged via the diode and discharged by the resistor when the PWM stops. The capacitor on the base will smooth the diode provided PWM so that you FET stays on during PWM. Fine tuning of cct required. Out of my head - probably OK but values may change.

The breadboard may be causing issues, check your layout (especially the feedback section)

Also, it's possible the inductor you are using is not suitable - it says it's only rated up to 100kHz, so it's SRF (self resonant frequency) is probably pretty low. It may be causing instability.
Try changing it to one with a higher SRF (e.g. >500kHz), but still with suitable current capability.

I did mention the output cap below but abdullah is right about the input cap being important. It does depend on the load, but the whole loop from in to out should be as small and low impedance as possible, ideally using a ground plane. On a breadboard that's "difficult" ;-)
If the frequency problem is not there with a steady load, I think as Kit says it's an output filtering issue, since the switcher won't be fast enough to adapt to high di/dt changes on the output and there's no "reserve". Increase the output filter capacitance and see if the ripple drops, if it does that's almost certainly the issue.

EDIT - Ah, I see you tried it with a resistor on the output.
In that case it would seem it's not the filtering. At this point I think I would use a different method of prototyping that's more suited to a switching regulator. Also use another chip just in case.
Either etch a board or use dead bug style, or stripboard with very careful attention to layout. If the frequency is still too high I would assume it's part of it's operation and not covered correctly in the datasheet - if this is the case then an e-mail to OnSemi is in order to see what they have to say.

EDIT 2 - Okay, after more reading I think the sense resistor (possibly combined with the inductor issue mentioned above) may be causing the current sense to trip too often and increase the timing capacitor charging slope. This will likely appear like the oscillator is switching faster.
A relevant quote from the App note:

When this voltage becomes greater than 330 mV, the current limit circuitry provides an additional current path to charge the timing capacitor CT. This causes it to rapidly reach the upper oscillator threshold, thereby shortening the time of output switch conduction and thus reducing the amount of energy stored in the inductor. This can be observed as an increase in the slope of the charging portion of the CT voltage waveform as shown in Figure 5.

Your oscillscope waveforms seem to agree with this description. Also, if you haven't tried changing the inductor, do this and see how it goes, plus you could try not using the current sense (i.e. just connect to input voltage)