Last winter I had the heater core replaced in my 2002 Liberty shortly after having a new raditor put in. Now this winter the heater core is getting clogged. I don't want to keep paying the dealer to unclog the core - how can I do this myself?

And why are some many jeeps failed here, named Liberty.
odd huh?
get rid of it.
but yes, you did not flush the whole system
or mixed coolants, or wrong, or too weak? not 50%
or slow head gasket leak causing filth in the core?
is coolant levels holding right, or dropping?

my guess is your post is all guesses.
why do i think that,?
why not post the symptom , both times, at rad fail.,
and now.
engine overheating?
engine under heating, not holding 180F+ ? (scan tool , ect, ah in 1min. flat the reading)
cores leaking both rad and heater?>
hose leaking,
miles on engine, 50k or 300k? what?

did you check the heater core for clogged or is that just wild guess.
you inspect it in car.
pull hose on engine side, and back flush it.
oops it not clogged, end guess;.
like poster alludes.

1: engine at 180f
2: both heater core hose at near 180f (IR pryro gun , 1min. work)
3: core damper open?

all my answers will be wrong, as always, because you never did say the symptoms, both times.

question 2:
are you commanding others to replace parts
or asking them to diagnose the unstated symptoms.

try this 4wd page on topic,
over heat and links to underheated engine.
http://www.fixkick.com/overheating.html

heater issues?
http://www.fixkick.com/AirCond/heater.html

The first mistake was fixing the heater core on the Liberty in the first place. If you can afford it get rid of the Liberty and get a new vehicle. I bought a 2002 new and it's been one repair after the next. Rocker arms falling off, serpentine belt goes about every 2 years, rear windows don't work and fall down, front suspension is shot, had the ball joints fixed via re-call. All in all, the heater core is best by-passed if you live in a warm climate, otherwise go for the nuclear option and replace the vehicle. In my Jeep it was blowing cold and when I filled up the overflow tank with antifreeze, it started smoking and the passenger compartment smelled like antifreeze and it overheated and cracked a head. After all this the rusting is unbelievable. Very cheaply made vehicle.

Core should not become obstructed that fast unless the engine has severe internal rust. You can remove the heater hoses at the engine and flush through the core (in both directions) using a garden hose. Likely though you have a problem with a blend door or actuator which unfortunately requires heater box removal (big job)
Hope you are lucky with that one and it does not need that.