Koga is the master of Poison-type Pokemon, and all of his Pokemon  save for one  are part
Poison-type. Poison-type Pokemon have only two weaknesses, and those weaknesses are
Psychic and Ground. Thankfully your Fire-type and Rock-type attacks will all be pretty
effective here as well.
The first Pokemon Koga sends out is his Ariados, which isn't much of a threat at all. The
worst it can do is Poison Jab for a fair amount of damage. Spider Web, which prevents you
from switching, is probably the most annoying thing Ariados can do, so make sure you
switch to the appropriate Pokemon to take it out on the first turn if it's not already
out, otherwise you may be unable to.
Forretress is a huge tank, being an unusual mix of Bug- and Steel-types and also boasting
massive Defense. Keep Physical-based attacks away from this bug thing and try roast it
with Fire-type attacks instead. If that doesn't work, something generic like Thunderbolt
or Surf  both Special-based attacks that aren't resisted by Steel  will take it out before
too long.
Muk is another pain as it can Minimize to raise its evasion, turning your attacks into a
coin flip against it. Try hit it with a strong Earthquake or Psychic to take it out
quickly before it has a chance to cripple you with its other disruptive attacks.
Don't let his Venomoth confuse you  though it looks like it's part-Flying-type, it's
actually just Bug/Poison-type, so don't go crazy with Thunderbolt or Avalanche just yet.
Though, since Bug is resistant to Ground, you might want to pick a different move other
than Earthquake, too. It's mostly annoying, though, so it shouldn't do much damage to
you.
Lastly comes his Crobat, which is very fast, has fair defensive stats and HP, and will try
Double Team to completely avoid your attacks. Now's the time to hit it with whatever
you've got to take it out! It has a lot of weaknesses: Psychic, Rock, Electric, and Ice,
and you're bound to have one of those types in your arsenal, so use that to take it out.
That should take care of Koga. Make sure you use Antidotes or Full Heals throughout your
fight to heal your Pokemon's status and also heal them after the fight so they're in top
condition for the third member of the Elite Four...