Many, many possibilities, but I'll try to hit the biggies...
First, make sure you have as many 3 and 4 stat modded clothing items as you can wear. Rings, necklaces, gloves, undershirt, etc - they all add up. Ideally, I go for con and stam mods (priority on stam for me, since I run out quickly).
Next, if you're factioned, keep tabs on when your side has a base under their control. Even the standard Stim-A's are useful, since that's a heal you can use while waiting for yours to recharge (small as that heal may be).
Third, buff, buff, buff!!! Elder buff, food, spice, and even get a medic buff if you know a medic that's not busy. The buff crystals can help too, but not if you do a lot of missions consecutively.
Fourth, if your target is wearing recon armor, consider using a Nym's. The kinetic dmg it puts out will make up for the the lower DPS (especially if they have energy layered recon - like 3000 kinetic resist).
These are the basics, regardless of your expertise temp. There are a LOT of different ideas on expertise - I have two BH's that I run different combos on, and I find I have a similar ratio of kills/losses on both. Personally, I think the assault armor bonuses and "shields" ability are mandatory for a good pvp BH - the rest is up to what works for you.
As for the actual fight, first, I NEVER have a BH-certed weapon equipped when I'm hunting my mark (ala GLC) - sometimes you can take your mark by surprise. I have my buffs and weapons on a seperate toolbar, and hit them all right before initiating the fight. I open with a snare and a bleed trap, then fall back and spam specials. And the bleed trap DOT is your best friend as a BH.
And lastly... don't use shields unless you're about to die. It can help you escape the fight and come back when your heal timer has reset.
That's all I can think of right now. I'm sure others here can give you their take on being a good BH.