How many pepper plants can I put in my half barrel? Also do I need to put a trellis around them?

I would just put one. You don't need a trellis for peppers.

I grow peppers in half whiskey barrels. I usually manage to put in at least 10 - 12 plants - I grow non-bell sweet peppers. I do stake my peppers as they tend to fall over in the wind. Cages work well. I plant the peppers in a circle and leave the center open. If you are treating them as annuals, why not crowd them?

I'd split the difference and plant half a dozen or so. I usually put 4 tomato or pepper plants in a shallow square storage bin, 6 in a big deep one, 2 pepper plants in a 5G ice cream bucket. These are in the greenhouse. I put one tomato in a wash-up pail, for easy carrying in outside and back in if the weather turns mean.

A half barrel is much larger. It takes 1.5 - 50qt bags of potting soil to fill my barrels. That's almost 19 gallons of soil. So, I can plant 4 times as many plants in my barrel as you are planting in your 5 gallon bucket. As I treat them as annuals, I can crowd them more and not worry about production. Did you look at my photo?

I use Miracle Gro potting soil and fertilize with Dynamite 10-10-10 half way through summer (in case that's the next question).

4X2=8. The big square bins are 18 G, and I'm only putting 6 in those. I could probably put more - not double, I don't think, and still get at them to harvest - but since it's indoors, I like to keep a little air-space between them, in case of aphids and white fly.

When I used half whiskey barrels, I put 4 peppers and three tomatoes per half barrel. Crowding more is ok if you're in a dry, fungal free climate. Here crowding plants in a container leads to all kinds of fungal disease.