Why do Klingons speak English to other Klingons?

The in-universe explanation is quite simple: they don't. They're speaking Klingon and the universal translator makes it sound like they're speaking English.

It's basically impossible to tell which characters speak English due to this device, as by the 24th century they're small enough to fit inside a Starfleet Combadge, as Captain Janeway mentions in The 37s:

Nogami: You are all speaking Japanese.

Jack Hayes: [to Nogami] Sounds to me like you're speaking English.

Janeway: It's because of a device we have, a universal translator. [points to her combadge] It allows us to talk to each other, even though our languages aren't the same.

The size of the device means it's basically guaranteed that their ship will have one installed for communicating with non-Klingons, especially as the Duras sisters from Generations have fallen out of favour with the Klingon Empire and basically survive by working with other species, such as the Romulans in TNG 5x01- Redemption II, Bajoran terrorists in DS9 1x03- Past Prologue and in Generations, as you mention, they're working with the El-Aurian Dr. Soran. As none of these species have any reason to speak Klingon or English, it's a fair assumption that they're all speaking their native languages and Lursa and B'Etor have a Universal Translator about their person.

Whether or not they're shown talking in their natural or translated language really depends on the director and writer. In some of the TOS movies, such as The Undiscovered Country, Klingons are shown to be speaking Klingonese whenever there are no humans around. In one scene, Kirk and McCoy are standing in a room full of Klingons speaking Klingonese, then we see there is a Klingon manually translating to English, before jumping back to the same actor speaking English, which indicates that the mouth movements matching up with the translation is simply a stylistic choice to avoid re-dubbing every time a Klingon appears to be speaking English in the movie.

I bet it's the same reason people from France speak English, just with French accents. The majority of English-speaking audiences don't want to read subtitles. Having actors speak English, but with accents, is a trope that filmmakers developed to simplify dealing with other languages.

(One fun example of this is Sean Connery's character in "The Hunt for Red October". He's supposed to be Russian. So for a few scenes, he speaks Russian. Then the camera trucks in on his mouth while he's speaking, so that his mouth fills the frame, and he starts speaking English with a Russian accent. He does that for the rest of the movie, if I recall correctly.)

Plus, as far as the Klingon language goes, it would be tough for normal actors to learn their lines.

For perspective, watch "The Hunt for Red October". Notice that aboard the Red October, at the start of the film, all dialog is in Russian. There is a scene in the captain's cabin where the characters are speaking Russian with English subtitles, and all of a sudden, they are speaking English and there are no more subtitles.

The point is that in-universe, they are still speaking Russian; the language switch is just for our (and the actors') convenience. The funny thing about Red October is that they did it in a way that is somehow subtle and obvious at the same time.

Getting back to Star Trek. There are two scenarios:

1. The Klingons (or for that matter, any other non-terrestrial race) are speaking amongst themselves and we the audience are the only humans listening. In this case, they are speaking Klingon (or whatever) in-universe; we hear English because we as the audience are privileged observers (and it's whole lot simpler production-wise) - the "universal translator" is hooked up to our TVs, if you must.
2. The Klingons (or whomever) are talking to humans. In this case, if everyone is speaking English, we must assume either they are speaking English (in-universe) or it's the magic of the universal translator at work between all parties.

There are scenes where the universal translator fails or there is not one present (or they intentionally don't use it). In this case, we hear Klingon.

Almost ALL aliens speak English, not just Klingons. I can't quote the names of the episodes, but I have watched the original series for years (own the DVDs) and remember most of the show plots. I remember episodes where Kirk and the crew went to planets and encountered species who didnt speak at all but used telepathy, and some other episodes where the script included a line or two that stated the aliens were of such higher intelligence they can translate speech so the humans can understand them. But I don't believe there was ever an episode in the original Star Trek that provided an overarching explanation for why everybody in the universe spoke English. I read on another internet forum that Roddenberry considered this problem and initially had a scenario planned for translation of alien speech, but this plan was discarded as too complicated (or something ) So the writers just let everyone speak English for their convenience and the convenience of American viewers. I assume they assumed we would just accept English speaking as part of the "miracle" of life in the space traveling future. And for the most part, we did. But it ALWAYS bugged me, just a little, even as a child watching reruns after school.

Klingon commanders/officers learn to speak English at a very young age. This is so the Command crew on board a Klingon ship can communicate sensitive or classified subjects between one another in front of the rest of the crew. It also comes in handy for monitoring Federation comms, as this is not widely known in Starfleet. Alternatively, the commanders simply do not want the crew to know who or what they are talking about certain subjects, thus a basic Klingon crew member does not learn English and will not understand what their commanders are saying when they chose to speak it. This comes from the Star Trek cannon universe explanation.