As a tourist I actually wouldn't bother getting to the market before 5am. Skip the tuna auction (there are a million videos on YouTube). The subways start running at 5, so catch one then and start with an early bite. Lots of places selling onigiri, find one. If you need caffeine, find a vending machine selling canned hot or cold coffee (often with Tommy Lee Jones staring at you) or try ordering from one of the small coffee shops in the market.

Wander the outer market—there are some interesting stores there. Head outside the market to Shin Ohashi Dori and find a food stall for breakfast. Depending on how much time you've killed, the inner market may be open or you can take a stroll over to Hama Rikyu Kien, a large garden next to the market. Head back after 10 and see the inner market. At this point it's mostly people cleaning up or taking a rest. There's still some business but it's mostly done for the day.

I think Hama Rikyu is better appreciated at the end of the experience, once you're tired enough and the little tea room is open. There is something indescribable about drinking tea in slow movements, looking out on the garden (or the bay) from the tatami room, resting your tired limbs, in a very quiet setting after several hours spent in crowded and noisy surroundings.

I went there almost 10 years ago (just me and wife, no kids) and it was an early highlight of my first trip to Japan. HRK itself is not the best garden, the tearoom is not the best tearoom, the tea is not the best tea... but after the hustle and bustle of Tsukiji, the HRK combo can be otherworldly.