How can I upgrade my circuit breaker from 15 amps to 20 amps? I recently installed recessed lighting in my kitchen, adding much needed light, but sometimes the lights flicker. If I reset the circuit breaker, it stays on for days at a time, then eventually it starts flickering again. Can I upgrade from 15 to 20 amps and how?

You can't upgrade the breaker without upgrading the wires and that may be difficult depending on how your home is wired. How many and what wattage bulb are you using? Flickering can be a bad connect or loose neutral anywhere between the breaker and the bulb. The breaker itself could also be bad. Lets start with finding out how many amps you are drawing with those new fixtures and what else is on that same circuit.
Just for your personal knowledge here is a wire size chart Wire sizes, gauges, for breakers (http://www.greatinspector.com/faq-elec-wiresize.html)

Agree with the loose connection, if it dims whan turning something on is different.
#12 is good for 20 amps, If it is #12, just change breaker to 20 Amp.
14 Gauge wire is safely protected by 15 Amp breaker. And Code

Do you live off a busy street? A loose nuetral in the panal with heavy traffic outside will cause lights to flicker. Sometimes all is needed is for you to tighten down the nuetral for that circuit in the panal.