A lot, I think a majority, of the deaths are skiers and ice climbers; There are also plenty of deaths from hypothermia, especially in the autumn.
There used to be a multi-decade list of fatalities for the northern presis available online, but I can't find it now. This page looks like it was cribbed from the one I remember: http://www.mountwashington.org/about/vi ... viving.php
Scroll down to see the various categories (hypothermia, drowning, fall, avalanche, etc) and keep a running tally by month.
In July your primary concern is heat and humidity. Do plan for wind and fog and temps below 60 Fahrenheit at the summit: bring windproof jacket and pants, a hat and gloves, and a light fleece jacket. But don't be surprised if you stay in T-shirts all day. Bring lots of water (or better, bring a filter and pick a route that stays near water most of the way).
In short, do NOT "pack along all my heavy mountain gear." You'll look like an utter pillock if you bring an ice axe in July, and the extra weight and bulk will actually diminish your safety.
I'm not a big fan of the Tuckerman Ravine route. Aside from being very crowded, you've got a long slog before a very steep headwall (admittedly with good ravine views and near some waterfalls) - I find that combination particularly tiring. It's also one of the more dangerous routes should you encounter slippery conditions or poor visibility. I suggest a loop up Ammonoosuc Ravine trail and down the Jewell trail. This lets you stop at the LoTC hut if above-treeline conditions are nasty (with option of getting Monroe instead / in addition to Washington), and on a good day gives you views to the west, south, and north rather than keeping you in an east-facing ravine until you reach the summit.