In other games, cheese strategies are usually pretty obvious. Early Roshan kills with Ursa in Dota 2, Starcraft Protoss hallucinate(all glory to HuK), etc.
That being said, in League of Legends we have a larger number of options to cheese with and while having more options, most of what I'll be highlighting will be very niche strategies so just make sure to take it with a grain of salt.
To give you an idea about what I'm talking about when it comes to cheese strategies, let's say the enemy team has an early game AD top-laner and an AD jungler. Not uncommon. For top lane, some people would just focus on hard counters or getting a safe or contested pick. Personally, I'd rather Cheez-It and pick Rammus.
Alright, we've got a little slice of cheese here but what's the big deal?
Well, assuming that you'll be mostly solo or ganked by the jungler, it's safe to assume armour is a good way to go. For a different flavour of cheese, let's cheese the runes and masteries as well.
A flat armour rune page will grant you 40 armour and grants Rammus 10 bonus AD. Flat armour rune page and straight defensive masteries will get you to 88 or so armour at level 1. Let's get some sharp cheddar in there as well with Rammus' Defensive Curl and a cloth armour start.
With this setup, you can have 130 armour at level 1 which grants you a bonus 32.45 AD(which totals at 86 level 1).
Now, assuming you're against a team that is 90% AD(with just the support in this example being AP), why wouldn't you pick Rammus? Thornmail and Sunfire Cape and then build AP from there with a Rylai's Crystal Scepter and a Zhonya's Hourglass.
Now, in most situations, this would be considered trolling. I mean who is going to have a flat armour rune page? I know I certainly don't.*I may be lying here. not my fault it's funny to watch people auto-attack me for like 10 damage at level 1*
The gouda news is that Rammus isn't the only one that can cheese. It just makes it easier that he already looks like a wheel.
Rush Athene's Unholy Grail on Zilean and get Tear of the Goddess and a Morellonomicon. 40% CDR and enough mana to spam your Rewind on cooldown. After using your ultimate, spam your Rewind and you'll have your ultimate again in about 38 seconds(assuming Rewind is max rank). Not bad for a 3 minute cooldown. Not the kind of hardcore cheese we're going for with Rammus but I'd still call that cheese.
Ever since Galio was released, he's been one of my favourite champions. I don't play him often but when I do, I cheese.
Flat AP rune page with a good offensive AP mastery page and a Doran's Ring will give Galio 77 AP at level 1 which gives him a bonus 46 damage on his Resolute Smite, totalling at 126 damage at rank one with a 24% slow and massive AoE.
End of the day, there are a large number of options that people don't usually look at because they're inefficient or not quite up to par with other things in a general game. That doesn't mean cheesing like Kraft is always bad! Most cheese builds are very niche to begin with so just be sure to keep that in mind.
TL;DR Grab yourself a salt cracker, cheese it up, and go ham, you'll make yourself a nice snack.