That looks cool already!
So, about the behavior thingy - I only see 2 ways of how people could use the interface. With a controller / keyboard, so that the controls are sorta like in every Zelda title or through the mouse, so it's probably a bit Diablo'ish. So, it'd have to be linked to the input system. Like newt already said, it'd make sense to have it in there as an object and as a behavior at once.
One idea is that we could set it to 'Grid' (typical keyboard / controller way like Zelda), 'Mouse' (like Diablo) or 'Combined' (so that both works and so that we can set up the controls accordingly).
I think it'd be the smartest thing to probably directly set up the appropriate settings the moment you use the inventory object in your scene. So, say for a Zelda-like inventory, if I pop up my inventory, I'd immediately expect that I could move along the grid using the same inputs that I'd use for moving my player around, right? So, Move Up / Down / Left / Right should directly be used without us having to setup control schemes. Otherwise, we'd have to create a whole Inventory Event Sheet with inputs like these:
if player presses 'Move up" - Inventory: Move up
If player presses 'Move right" - Inventroy: Move right
Which still could be nice for whatever as an option. So we could override the defaults, but I think 90% of the users would settle for a Zelda or Diablo like Control Scheme. If I look at all the Inventory systems out there, they're all either like Zelda / Final Fantasy / Diablo, really.
Also, if we pop up the inventory, as long as the inventory is active, it'd probably be smart to give the user the option to freeze any inputs on other layouts. Imagine a Zelda game where you're running around on a cliff and quickly need to switch to an item, you pop up the inventory, press 'Move Right' 3 times and boom, you fall down the cliff, but at least you're at the right item spot in your inventory. So, if Inventory is active: Ignore all user input on all other layouts.
Same thing with the mouse. You'd want to have stuff in there like: If mouse hovers over an item spot, do whatever. Like, change the backdrop, add an effect to the item spot or something, so that you'd always get an input what you're currently hovering over. And the option for: if inventory is active: Ignore Mouse input on all other layouts.
Then we'd still have to setup how we'd control the inventory, cause that'd probably be a little different for everybody:
If player presses 'Tab' - Inventory: Show Inventory
If inventory is active + If player presses 'Tab' - Inventory: Hide Inventory
And now it's about interacting with the inventory:
If player presses 'Action' + Inventory is active - Inventory: Use item
For keyboard only. Or:
If player presses 'Action' OR if player right clicks + Inventory is active - Inventory: Use item.
Then we'd still have to find out what kinda options we'd need.
Use Item? Drop Item? Equip Item? etc.
Now that I'm thinking about it, it'd also make a LOT of sense to have a 'Menu' Object too, right?