There's no option for that.
As far as I know, there's only Tavern Brawler that can give you proficiency with improvised weapons.
But it doesn't make sense to be improvised.
Improvised weapons are pretty much defined by having no proficiency bonus (tavern brawler aside), so there's no real distinction between using a weapon you're not proficient with and using a weapon-like improvised weapon such as a pitchfork 'trident'. So there doesn't seem to be much point to invoking the Improvised rule.
It could just be a re-skin.
The easiest way is to just treat a spiked chain as a variation of one of the standard weapons, in much the same way that the monk's Martial Arts section (PHB p.78) discusses treating nunchaku as a special kind of club and so on. Treating a spiked chain as mechanically identical to a whip, pike, or halberd would probably be the most direct solution.
A kensei monk (Xanathar, p.34) might pick 'whip' as his kensei weapon, and just describe it as a spiked chain/rope dart/etc. You might even describe how, as you level up, you're moving to heavier and more elaborate chain weapons, which are all just a 'whip' from the mechanical side, and are technically getting increased damage dice from your Martial Arts feature rather than from being more dangerous weapons.
However, I suspect what you really want is something like a whip that deals better damage without using the kensei's Martial Arts. Mixing finesse with reach and good damage would be a pretty powerful combination, especially for a Rogue, so you would certainly need some restrictions to balance it.
How could you handle actual Exotic proficiency?
Since the core rules only have 'simple' and 'martial' weapons and only a very limited number of ways to gain a proficiency, there are only a few ways to address something like an 'exotic' weapon.
The Weapon Master feat would certainly make sense in this case; or you could home-brew a feat that gives you proficiency with one 'exotic' weapon and some other side bonuses. Probably the best way would be to think of the feat as adding damage with a specific kind of weapon -- in this case, the proficiency would be a lot like a feat that says "when you hit with a whip attack, you deal +2 damage" or some such thing (if the Spiked Chain were a d8 reach finesse weapon, for example).
But there are problems with that.
It might be worth-while for an "exotic proficiency" feat to require you already be proficient with all simple and martial weapons as a prerequisite, since otherwise exotic weapons immediately become the go-to choice for a character who isn't already omni-proficient and wants to pick up better weapon. (That is to say, a paladin is already proficient with everything, so a feat that lets him pick a slightly superior version of one of those weapons represents 1 step up, while a wizard or cleric who acquires an exotic weapon proficiency is jumping up 2 steps.)
But that doesn't fix everything. There are ways to get around the proficiency system -- the warlock's pact blade, in particular, makes him always proficient regardless of form, so all blade-pact warlocks immediately find exotic weapons very attractive. Since the game doesn't have exotic weapons as a standard element, the rules won't contemplate them, and you have to be ready for unusual rules interactions. You'll need buy-in from everyone at the table, and an understanding that the DM can nix any weird combo effects that might show up.