All the random stuff is based off the gear you are wearing. So you can buy or just put on higher gear from another character, and you'll supposedly get stuff around that.
Hasn't been working for me as far as drops go. The renown boxes seem to drop near or slightly better.
Fastest way to increase your item level is to spam flashpoints. The gear dropped from bosses and looted from the bonus boxes will be +-2 ilevels of your current ilevel. I would estimate upgrades of +2 happening ~60% of the time, up till items turned blue, then it might have dropped to ~30-40% for a +2. There might be a chance at a +4 but if it happened, it was much more rare, maybe 5% or less.
To go from the full set of 268s (I think it was) you get from completing the Onslaught Story to 306 took an estimated 60-100 flashpoints (with opening most boxes). After getting all 306 on my Jug, I built another set for for a second toon, and it was getting 302 to 306s right off the bat, in flashpoints. Random drops from the world do not scale up to 306, btw.
In most cases, as far as keeping or using the boxes go, it can go either way, and all depends on your end goals. The bonus boxes you get from doing a random flashpoint can have up to 4 items in them, any of which can be a set item or one of the new tactical items (very low percent chance). I have received 60% of my free set items from FP bonus boxes that I opened after hitting ilevel 300 (or whenever items go from blue to yellow quality). Buying set items from the vendor cost 1 million credits and 3000 of the currency (that you get from breaking down level 75 gear or doing some quests). As a reference, it takes about 5 flashpoints to get 3000 currency, when disassembling almost all the loot.
Saving all your boxes from randoms would drastically slow down your climb to 306, but increase your chances at free set items, and faster acquisition of separate 306 gear for alts (although you can pass around a single set of 306 gear if need be). There are probably 25+ sets, so the chances of getting what you need aren't very high.
There is one box that should probably be saved till 306, and that is the weekly conquest box, if you received it after hitting 75. It has had a much higher chance of dropping a set item.
If you are going to be playing for a while, also think about joining one of the top guilds on the conquest leaderboard, as they get more of the conquest boxes, as well as bonus tradeskill items.
That said, I also read the recommendation that boxes should be saved, so I started to save all my boxes at first, but it was too slow. So, I eventually started to open all I had until I got to around 300. After that, I started saving renowned boxes, which was probably pointless, as bonus FP boxes would have been better to save. I have had to buy 16 of the 18 set items I am using, but I have about 20 useless ones sitting my my bank. On the other hand, all my tactical have all come from drops or boxes (about 10 in total), 3 of which were BiS.
