Planning an extension to wardrobe over the stairs, but...

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Well I'm a bit green.
In an insomnia-induced binge of info I've figured out that I need to be sure to leave 2m headspace over each step, that's easy enough to plan around but the designs I've seen differ a little from the plans I had drawn up and now I'm questioning the stability of my own ideas. We have an old boiler cupboard in our master bedroom that sits "overlooking" the stair well. 2 of the walls are brick, one is stud wall. I've been thinking of busting the stairwell open and making a box in the side of our cupboard (Which was a shower when we moved in.. That was fun to rip out.) so I can try to find somewhere to hang my clothes when the SO has put all hers up on rails.

I had planned on taking out the wall about a foot up from the floor and to the ceiling, and a frame for the floor held up by the rear, brick wall and the stud wall overlooking the stair well (Cutting out a rectangle of plaster to squeeze the frame in) but.. While I figured we'd never step on it, so I wasn't too worried about how much weight it could hold, I keep seeing that people brace these frame from beneath with angled joists.. So I'm thinking that'd be a much wiser design choice..
Guess I'm just looking for any tips. Been renovating my first house since May and so far it's gone really well by merit of determination and common sense.

My drawing is not particularly to scale but it's a good approximation.
 

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I cant fully make your drawing out but would put joists on hangers between the brick walls, similar to a floor spec, to make a safe base for the cupboard, and go from there.

Blup
 
You say you won't step on the base, but others, such as children or new 'occupiers' may.
You really do need to support the base in some manner.
 
support the base by 4x2 flat at the back and chamfer the underneath to the stair angle [up to 43%] then as you go forward 4x2 vertical
the stairs wont be 45% but to keep it simple go for 45% and you wont get confused unless off course its a far shallower angle then you choose
 
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