Given how many writers have been involved with Doctor Who and how long it's been running, it's fairly common for stuff like this to be rather inconsistent.
That being said, the Doctor can at minimum recognize other languages, because in "The Girl in the Fireplace," he knew not only that Madame de Pompadour was speaking French but what period French she was speaking. So, unless the TARDIS is telling him that in addition to translating, then he's fluent enough in French to recognize dialects from different time periods. And given how often he's dealing with English speakers, if he's fluent in French, he's bound to be fluent in English.
Also, in "The Sontaran Stratagem," he was correcting Luke Rattigan's grammar, which would definitely indicate that he's fluent in English (and probably more fluent than your average native English speaker given how bad many people's grammar is).
On rare occasions, he appears to have spoken directly in an alien's language, and the companion doesn't understand either him or the alien (e.g. "The Planet of the Dead" with the fly people), which would indicate that he's fluent in their language. The companion may not have understood in this particular case, because the TARDIS was on another planet, but the lack of the TARDIS hasn't stopped companions from understanding in the past either, and I'm pretty sure that there have been other occasions where the Doctor has spoken a particular tongue without the TARDIS' help in the old series if not the new. In either case, the Doctor definitely speaks other languages without the TARDIS' aid from time to time.
On even rarer occasions, he hasn't understood something (e.g. he didn't understand the writing on the wall in "The Impossible Planet" because the TARDIS couldn't translate it), so clearly the TARDIS is doing some translating for him and not just his companions, but I think that it's fairly clear that he's fluent in English as well as other languages, though who knows how many.
But again, it's not like this sort of thing is consistent. The first time that automatic translation was even mentioned was with the 4th Doctor, and even then it was a "gift of the Timelords" and the TARDIS wasn't mentioned at all. It wasn't until the spin off novels that it was suggested that it was the TARDIS which did it (something which they carried over into the new series). Prior Doctors have had occasions where they didn't understand languages or writing (rare as it's been). It's just that it was more convenient for everyone involved to have all of the characters speaking English. It wasn't until much later that they felt the need to explain it, and as with many Doctor Who explanations, the explanation is ripe for being contradicted by future episodes if it's not already contradicted by previous episodes. As great as Doctor Who is, it's not the type of series that has consistent in-universe explanations for things. It's just been going too long with too many people involved, many of whom clearly didn't necessarily care all that much about continuity.