I did a hose installation alongside a professional mechanic yesterday - new but partially drained components on an upright road bike.
We did:
0) Check lever adjustment is set to max, and calliper pistons are appropriately positioned. (We did this part way through.)
1) Bike angled so the hole in the hoods is horizontal, with a slightly-filled reservoir fitted.
2) Feed from a syringe at the calliper, until bubbles mostly stop arriving at the reservoir.
3) Swap out syringe for a drainage bottle (remember to bung and tighten the relevant orifices!).
4) Make sure there's enough oil in the reservoir and allow system to drain into the bottle until no bubbles.
5) Tip bike back so hoods at 45 degrees, frob levers and open and close bleed port (if we're thinking of the same Shimono doc, I think it's detailed here), keep going until you get pressure on the levers and bubbles have gone. Repeat with bike tipped forward 45 degrees.
6) Repeat for back brake. (We tipped the bike up loads to get a downhill run on the hose, then rotated the bars to get the right angle at the hoods. Have fun doing the equivalent with a bent!
The mechanic was also saying they'd ideally then leave the bike hanging up for a bit to allow any remaining bubbles time to work up through the system.
Isopropyl alcohol spray and lots of tissue and jay cloths!
All the best with it.