It depends on who you're talking to. If you are looking at a formal guideline it'll say it's not correct, but those rules are based largely off of Latin rules which don't really apply to a Germanic language very well. Further, they are increasingly becoming out dated. I shouldn't have said it's "incorrect" because it's not really incorrect but if you wrote that in a school paper your teacher is likely to put a red line through it.
Churchill was a little confused with "...up with which I will not put". 'up' is not a preposition in that context, it's an adverb or particle (I've see it called both), so "This is the sort of English with which I will not put up" would be the strictly grammatically correct version of your sentence and far more natural-sounding.
Yes, it is definitely a double negative - we actually use double negatives all the time in English although we're not supposed to, or rather we learned that we weren't supposed to (we're also not supposed to end sentences in prepositions like I just did). I'm not saying it's wrong to use them but technically speaking its incorrect if you abide by traditional or standard English grammar rules.
@heimaey No, "it's not unusual" does not mean "it's not usual". "It's not unusual" builds a positive sentence ("it's usual"), but in a less strong sense (i.e. weaker) than actually saying "it's usual". This case of the double negative was never even seen until 1762 - the 1700s. The example on the link I provided ("not unconvinced") is a direct equivalent to this one ("not" + "un" prefixed to an adjective).
That link just proved what I'm saying - standard English doesn't allow it. It's not unusual doesn't fall with in the exception you linked to - in correct standard English grammar it should be: It's not usual, but in every day English we use them - and we used to use them all the time until around the 1600s they got in in their head that it was wrong for whatever reason.
I can't reply to your other comment for some reason so I'll reply to this. I am not going to agree with you because you're wrong - in written standard english double negatives are a no-no. Look at any grammar book. However, in reality we use them all the time and these rules are likely to change just like we are kind phasing out whom when who is perfectly acceptable. I'm done here because you keep telling me that double negatives are fine and I agree except when you consult a grammar book and then it'll tell you no - and probably if you're in an English class they'll tell you no as well and give you a bad grade. Done.