I have written on Stalin's language abilities before (see here, but will repost it here with some additional information:

Stalin indeed was noted for his Georgian-accented Russian, as he was ethnically Georgian and grew up in Georgia. His initial writings were also exclusively Georgian, and he didn't switch to Russian until 1907, in part because his Georgian features made him stand out amongst the Bolsheviks, which while beneficial at times (it was a factor in him becoming the authority on nationality policy), he wanted to fit in more, and Russian would have a far wider audience than Georgian.

And for what it's worth, Stephen Kotkin's Stalin, Volume 1: Paradox of Power, 1878-1928 (2014) provides some information on the languages of Stalin, which I'll quote here (p. 10):

...Georgia was a diverse land and the future Stalin picked up colloquial Armenian. He also dabbled in Esperanto (the constructed internationalist language), studied but never mastered German (the native tongue of the left), and tackled Plato in Greek. Above all, he became fluent in the imperial language: Russian.

Donald Rayfield, also wrote on Stalin’s language abilities in his 2005 book Stalin and his Hangmen: The Tyrant and Those Who Killed For Him (p. 22):

All that hampered Stalin were his linguistic limitations: only in Georgian and Russian could he cope without a dictionary. Yet here too Stalin was underestimated by his opponents. In the seminary he had learned a lot of Greek (visitors were amazed to find Stalin in his Kemlin office perusing Plato on the original) and afterward a little French and German. For a while, in Siberian exile, he even toyed with Esperanto. Stalin’s interest in Marxism and his first prolonged stay in Berlin impelled him to struggle with German periodicals.

People wrote to Stalin not only in Russian and Georgian, but also, from Baku, in Azeri Turkish (then written in Arabic script). When on the run from the police, Stalin sometimes went under the name Zakhariants or Melikiants; either would have been foolish without a smattering of colloquial Armenian. In 1926 during the British General Strike, and afterward, Stalin perused the British press. His letters to his wife from Sochi express annoyance at her forgetting to send him his copy of A Model Complete Teach-Yourself English Course. In languages, as many other subjects, Stalin’s tactics were to conceal, not display, his knowledge.

Now for some context on the above:

• Armenians were the dominant people in Tbilisi, the capital of Georgia, and ran the business interests of the city, and of Georgia as a whole. As Stalin worked (albeit briefly) in a shoe factory (owned by Armenians, like nearly all factories), it is not surprising he picked that up.

• Esperanto: This is the first I heard he knew any of that, but its not surprising, considering the purpose of the language (a common language for people) and its interactions with leftist movements.

• German is expected, both because of the association with Marx, but also because Stalin lived in Vienna for some time, and dealt with German Marxists frequently.

• Learning Greek at a religious seminary should not be surprising, especially when it was a Georgian Orthodox seminary. While the Georgian church uses Georgian in their services, the historic ties of Orthodoxy and Greece (and the Greek language), are strong enough. It would be analogous to learning Latin in a Catholic seminary. There was also a significant Greek minority in parts of Georgia, and while their numbers have decreased substantially since then, they still exist there today. Considering Stalin's frequent travels across Georgia and his mingling with the people in the country, and his reading habits (he would read hundreds of pages a day at times, especially history), again it is not unexpected.

• Azerbaijan borders Georgia, and at the turn of the century many ethnic Azeris (especially from northern Persia) went to Baku to work as labourers in the oil fields. They were prime targets for Bolshevik propaganda, and with Stalin spending time in Baku it would be a fair assumption to expect him to learn some of the language. Rayfield also notes that it was written in Arabic script at this time: Azerbaijani was given a Latin-based script in 1929, which was replaced by Cyrillic in 1938. After regaining independence in 1991 Azerbaijan again adopted a Latin script for its language.

• English was starting to become an international language at this time, and as Rayfield notes Stalin took active efforts to learn it.

• Russian was obviously the language of governance and the Bolshevik movement as a whole. I don't think this needs to be explained.

To try and answer the question now, it is important to note that at his conferences with Churchill and Roosevelt during the Second World War (and at his meetings with their ministers in Moscow throughout), Stalin always relied on a translator, and never spoke English with them. Whether this means Stalin didn’t understand English, or preferred to rely on a translator, is thus difficult to say with any certainty, though based on Rayfield’s statement I’d be inclined to lean towards no, Stalin did not speak English well (I will also note that Rayfield is a former professor of Russian and Georgian, so I would trust his analysis on language abilities).

This is not unusual either, and has a parallel today: Vladimir Putin is well-known for being fluent in German (he worked in East Germany for the KGB during the 1980s), and is apparently conversational in English, but always uses a translator when dealing with foreign leaders. This is more interesting because Angela Merkel, the Chancellor of Germany, grew up in East Germany and is herself fluent in Russian. So both Putin and Merkel can speak to each other in two languages, but use translators.