Solution 1:

OED provides no help, only an early citation:

French letter n. colloq. = condom n.
?1844 Exquisite in P. Fryer Man of Pleasure's Companion 131 Gentlemen who live in London will be at no loss in easily obtaining these ‘French Letters’.

Etymonline gives some help and a plausible explanation:

French letter "condom" (c.1856), French (v.) "perform oral sex on" (c.1917) and French kiss (1923) all probably stem from the Anglo-Saxon equation of Gallic culture and sexual sophistication, a sense first recorded 1749 in French novel.

I suspect that it's called a letter because of its packaging at the time, but I don't know how to verify that.

Solution 2:

I was sure this was a duplicate, but can't find it.

Many countries with longstanding rivalries use each other's names to mean "fake". For example, in the UK:

  • French windows are actually doors (called French doors in the US)
  • French leave is going awol (in France they say filer a l'anglaise for the same thing)
  • French letters are condoms

While in the US:

  • Dutch courage is being drunk
  • a Dutch treat is neither treating the other
  • a Dutch uncle is not your uncle
  • a Dutch oven is not an oven (it's a heavy lidded pot you can achieve baking-like results with on the stovetop)

and so on

As far as I can tell, in the US, French typically means "cut into long thin strips" (French fries, frenched beans, frenched ribs on a roast) or "the glamourous luxurious way they do it in France" - French vanilla, French bread. In some parts of the country it once seems to have meant racy or sexy - the French postcards in Oklahoma! for example. I think French kiss falls into that category.