As a mechanic, I would suggest that using a garden hose, and back flush from the fan side out, and use an appropriately sized zip tie to help move the more stubborn bits of insect and baked on chips/ dust etc.
take your time and be careful/ gentle if its been a long time since done.
Most times, just a careful blast of air, or back flush with hose water is enough.
As the fins, (the fins between the rods that hold the coolant fluid) begin to age and perish, be mindful of it, and if you want to extend its life a bit, get a rustguard type black paint, and take rad out, clean, and hit with a few cans to help seal up the now corroding copper fins, makes them last a bit longer, or if done early enough, make it last a lot longer, providing your using the right corrosion inhibitor and changing it out as required, as it dosent last forever.
In saying all this, ive seen people use too much pressure, and pretty much ruin the core that could have lasted much longer if care and a bit of time was taken.