Quad builds are usually pretty dependent on a +Spd nature. Honestly though, as long as he's not -Atk you can pretty much do anything with him. Klein is one of the lucky few who has a great set of base skills already, so you really only have to worry about deciding on a support skill and a C slot, which are all mostly team oriented.
Honestly Klein has a fantastic standard kit that only comes to second after Takumi w/o inheritance. Only unfortunate thing is that +atk IV is pretty important to him since it effectively becomes +6 atk due to brave bow attacking twice. Death blow is a great A skill to keep since it effectively becomes +12 atk when factoring brave bow. The best bet with B skill would be swordbreaker or lancebreaker, which will ensure that you kill virtually all sword/lance users that are otherwise too fast or tanky for Klein to kill normally.
If you want to go the Quad route, I think Jeorge or Takumi is better at the job due to higher atk/spd total
I planed to make a Quad Takumi with the one I pulled in the gold banner but then I pulled a Klein today +SPD/-DEF and tried him with the same kit (L+D, Swordbreaker, Luna, +1 ATK Seal) in the damage calculator just for fun... it turned out that he is even better than my Quad Takumi with 112 OHKOs...BUT he looses to Hector no matter what and to fury or +SPD Ike and Ryoma which are very common. So Klein could be build as Quad Klein but I think he has to be +SPD or +ATK at least to do the job well.
For your Klein the standard build seams to be good. With Luna and Swordbreaker as Joonie said he can even take out Ryoma, Ike, Lucina, Zephiel and Xander which is great!
Klein is one of those heroes where you don't have to do much inheriting, His default build is amazing. Slap threaten defense on him and you are good.
Other options are breakers to better counter a unit if they are giving your team trouble. I use sword breaker on him sometimes to counter Bulky Red units, specially Ike.