Yes you can still be good without using the battle rifle here are some tips that have worked for me when i sucked:
1 Always go for the big powerful weapons: rocket launcher, shotgun, sword, brute shot, sniper rifle (even if you are not good so the enemy won't use it against you)
2 Always stay inside, keep moving but try to stay inside all the time.
3 The assault rifle is one of the best weapons that's why they bring it back from Halo 1, never discard this weapon.
4 Always run towards the enemy you want to kill, if you start going back or trying to escape you will most certainly die.
5 When you want to kill someone try jumping over him you will have advantage.
6 As i say in 4 when you run towards him keep shooting no matter what weapon you are using until he is in range so you can melee him.
7 Always try to go for the camo and overshield.
8 Spare no expense on grenades always throw grenades, specially when you are about to die you might get lucky and get kill from the grave.
9 Go for the machine gun and vehicles (the only reason to go outside)you most certainly die but not without killing at least 2 guys.
10 Don't use double weapons they don't let you use grenades and most combinations suck.
11 When you play on team doubles, team slayer, multi team, etc. Always stay with someone else never go solo.
12 Know the map you are playing in, where is every item, where you can hide an assassinate someone.
13 Adjust your sensibility so you can move fast but still able to control the, turn the vibration off it distracts you and makes you less accurate.
14 Remember to use your equipment (bubble shield, power drain, gravity lift, etc)
15 An to finish this as any sport, science or hobby the most important thing to to is PRACTICE, try playing at least one hour a day and even though you can still be good without using the battle rifle it is still one of the best and most easy to find weapons so try practicing with this weapon, also these are the best weapons on the game try practicing with this also: shotgun, rocket launcher, assault rifle, sniper, brute shot, mauler(brute pistol), carbine, plasma grenade so you can stick it to them.
Ps: This is just a theory of mine, but i like using the elite since i think is harder for other players to locate the head.
EDIT
I don't know why someone gave Bisrtg a thumbs down but what he says are also very good tips.