"He's the student I'm teaching English this year" perhaps?

It's one of those JPN-ENG pairs, so I didn't want to change it without having some Japanese-knowing person take a look first...

The 'minimum change' would be

"He is a student who I'm teaching English to this year."
(I think the 'to' might be able to be omitted, but that sounds a little odd to me)

It isn't clear whether he is 'a student' or 'the student'. On logical basis I'm assuming that nobody can survive on payment for teaching just one student so it's probably 'a'. ;-)

The 'who' could be omitted (and it probably sounds more natural that way). It's not 'wrong', though.

"He is a student to whom I'm teaching English this year"

This brings "to" to the front and thus makes it more natural, IMHO.

> This brings "to" to the front and thus makes it more natural, IMHO.

Not bad, but you lose points because nobody under 25 uses 'whom' anymore. ;-)

Seriously, we can put that in as an alternative.

Depends on where you grew up and where you live. I can assure you I am squarely under 25 and use "whom" a lot.