As far as I know, there are not actually any stats for the katana in 5E yet. So you can give it quarterstaff stats with damage changed to slashing, and declare it a monk weapon.In terms of making a character within the rules that fits that vision, one possibility is
Character class: Monk
Chosen weapon: Quarterstaff (1d6, 1d8 with 2H) reflavoured as a katana doing slashing instead of bludgeoning damage
Then he can have fun being all light armoured and wise, quick and athletic and do his versatile 'quarterstaff' fighting with two hands, getting his bonus unarmed attacks and other stuff. Uses Dex instead of Str for attack and damage.
I'm sorta naturally suspicious of anyone who wants their non-Oriental Adventures character to wield a katana, but that's neither here nor there.
Unfortunately, without houseruling your player might be SOL. The biggest problem seems to be the lack of a two-handed finesse weapon; if you decide to make one, I recommend that it should deal 2d4 slashing damage at most.
That's a great idea.
Take the defensive duelist feat and go the emental monk path and you can be pretty bad ass.
at 5th level with the fangs of the fire snake (refluffed to more cool katana strikes) you can have something like:
AC 17 (20 with a parry reaction)
Two attacks at +7 1d8+4
Bonus attack at +7 1d6 +4 (no monks get there abil mod on bonus attacks)
and the ability to use a Ki point for a 2nd bonus attack AND/OR use another two to up the damage to 1d6+1d10 with a range of 10'.
I would blow thorugh Ki points like a 9 yearold with a bag of skittles....
A bit late to the thread, but I'll say that the suggestion above of using a Monk + reskinned Quarterstaff is a stroke of brilliance.

(Dungeons & Dragons)
Rulebook featuring "high magic" options, including a host of new spells.