Well, if you don't mind making a pact with some powerful entity, you could get 2 levels of warlock. This will give you access to a few spells, some of which perhaps being interesting (both stats and flavor-wise) for your character. But most of all, this will give you access to 2 invocations.
One of these invocations should be Devil's Sight, which grants you the ability to see through any form of darkness, magical or otherwise, for up to 120 feet. This is more than enough to cover the 15 feet of the darkness spell. Cast it on your clothes, and bring terror to your enemies.
While this is a good strategy (generally favored by blade pact warlocks), it has some limitations. While under the effect of the darkness spell, your allies cannot see you. As such, any spell requiring line of sight that they would like to cast on you simply cannot work. Healing you might be problematic, as it requires your allies to be able to touch you. They might know that you are smack in the middle of that sphere of pure black darkness but, once they get inside, it's suddenly not so easy to find you anymore.
These limitations also affect your enemies, however. Any attack from the outside of the sphere should have disadvantage as your enemies, even if they can approximate your position, can't exactly see you to aim properly. Enemies inside the darkness should also suffer from disadvantage when attacking you, unless they possess some form of blindsight or a similar ability. When you attack them, you should have advantage on the roll since you are functionally invisible for them.
The biggest cost of this strategy is the fact that it requires multi-classing. While the lvl 19 and 20 abilities for monk would not really be missed (realistically, few campaigns will reach these levels), it will still set you back 2 levels in obtaining you other core monk powers. Note that doing this would get you 2 invocations, however, and that some of them can be quite interesting for a ninja-esque character:
Armor of Shadows (cast mage armor at will, which might be better than
using your wisdom for AC if its under 16),
Eldritch Sight (cast detect magic at will),
- Eyes of the Runekeeper (allows you to read anything, even languages
you don't know... useful for spying!),
- Gaze of Two Minds (share the eyes of a willing target, more spying!),
- Mask of Many Faces (disguise self at will!), etc...
TL;DR
Sacrifice 2 monk levels and multiclass as a warlock. Gain some minor but potentially useful spellcasting, as well as the ability to see through magical darkness and one more ability of your choice!
Smoke grenades
In Chapter 9 of the DMG, explosives are introduced as option for the DM to allow in their game. One such explosive is the smoke grenade which is completely mundane and creates a heavily obscured area:
One round after a smoke grenade lands, it emits a cloud of smoke that creates a heavily obscured area in a 20-foot radius.
This seems to be the only concrete case of a defined, non-magical technique to create a heavily obscured area currently in the rules.
Note that this is an optional thing that cannot be assumed to exist in every game. Ask your DM if they'll allow it (or some modification of the idea).
Unearthed Arcana: Smoke Stick
There is one option outside the currently released rules in the Alchemist subclass of the Artificer Unearthed Arcana class.
Smoke Stick. As an action, you can reach into your Alchemist’s Satchel and pull out a stick that produces a thick plume of smoke. You can hold on to the stick or throw it to a point up to 30 feet away as part of the action used to produce it. The
area in a 10-foot radius around the stick is filled with thick smoke that blocks vision, including
darkvision. The stick and smoke persist for 1
minute and then disappear. After using this formula, you can’t do so again for 1 minute.
This is not a piece of equipment though, but a class feature. However, it does provide another rule option (albeit in playtest form) that indicates how creating non-magical smoke would/could work.
Best Answer
Blindsight or Tremorsense are the only abilities that let you 'see' through fog or smoke.
I'll spare you my original overly-off-topic explanation; suffice to say that according to the rules of visibility in 5E, when something is blocked by something else, there is nothing anyone can do to see through it except for possess the above abilities. You are 'blinded' by the object that is blocking you, and until that thing that is blocking you is removed, you will always be 'blinded' by it.
Tremorsense and Blindsight are the two things that RAW let you detect the location of things without having to see them, and as such these are the two RAW things that would let you look through heavy smoke. Neither is available as a class ability, so your PCs are not likely to have access to them.
As you mention, Blindsense does not let you see through smoke, because Blindsense only works when the person you're looking for is hidden (took the hide action) or invisible (had invisibility cast on them).
If you're just standing in a fog cloud, then someone with Blindsense cannot detect you any better than someone without Blindsense.
If you're worried about a smoky room, you can look at the spell Fog Cloud to see other ways to get around it, mainly:
So, it looks like RAW, mild to moderate winds can disperse fog/smoke to a degree that it stops interfering with vision. In that case, if one of your PCs has a spell that generates wind, then by the Fog Cloud rules they could use it to disperse the smoke and see through it.
They could also have the party barbarian tie large paper fans to his arms and flap them like the dickens, but that may or may not be too silly for your campaign.