I used to do something similar- so I speak from experience when I say to cool the wort a bit first by sticking in in the sink in a ice/water bath. You'd think that adding 3 cold gallons to boiling wort would cool it fast, but it doesn't. It seems to get to 120-130 pretty fast, and then take forever to get cooler. This is the danger zone for bacteria. And it's harder to cool 5 gallons than 2!
So, cool the wort first, to under 100 degrees. THEN add the cold water and the boiled wort to the primary. I'd add the cold water a gallon at a time, just in case it cools it too fast, then you can add lukewarm water. Once you do this a few times, it works really well and you know what temperature your wort needs to be to be cooled to 70 degrees with the cold water.