Early July-hmmm. The places in northern Oregon where you're likely to run into lots of unmelted snow (enough to make it hard to find the trail, probably also requiring traction devices and an ice axe) and dangerous stream fords due to high water are on the side of Mt. Hood, the Mt. Jefferson Wilderness and the Three Sisters Wilderness. Otherwise you'll probably be fine. Those three areas usually don't melt out until late July or sometimes into August.
A few years ago I actually won a contest (no prizes, just fun) for guessing the date that snow would be completely gone from the trail at Paradise Park (where the PCT goes along Mt. Hood). My guess (actually a completely wild one) was August 2. Of course the trail was useable 2-3 weeks before that.
While our snowpack in northern Oregon is again above normal (though not as much as last year), we have had a slightly warmer spring so the meltout is close to normal. Everything depends on the weather between now and when you leave. Right now the forecast is for cool and rainy weather for the next week. It could stay that way (with more snow at higher elevations, as happened last year) or we could get a heat wave. Hope for the latter!
As Brian stated, www.portlandhikers.org is a great source of information. We just had an inquiry (from Spain!) about backpacking the PCT from Cascade Locks to Timberline Lodge at the beginning of July. We gave the same answer as above. That thread is here:
Keep watching the trip reports on that forum. Also, this time every year we usually have a snow level thread that people watch anxiously.
Another place to watch is this snotel, on Mt. Hood near the PCT at 5,370 feet:
Note that it uses Snow Water Equivalent in inches, rather than snow. At this time of year, double the inches of Snow Water Equivalent will give you the approximate inches of snow. The great thing about this particular graph format is that it shows you the current year, last year (so you can see how horrific last year was) and the 30-year average. You can see from that that it might be better if you could wait until July 15 to start.