You need strong characters, gold especially. If you don’t have a lot of strong characters then wait for them to regain energy before contiuing the PF. Using weaker characters at high streaks will just make you lose and reset.
It may help to have a team of 2 strong characters + 1 weak character to help build up your streak, but once you get around 10 and up that won’t work as well anymore.
Also, try to refrain from fighting turtle teams, like 3 Bella’s or a team with Resonant Evil Big Band.
Aeon or Venus--AI teams that are randomly generated, if you see one of these pick them, they're like candy.
Dash canceling--tapping after a dash is tricky to time but will net you a safer approach and a free combo sometimes. Often you can see the enemy rearing back for a punch that would've connected with the slight delay a dash attack has just before it hits. With a dash cancel you get rid of this open frame.
Study combos in the training room--often a long combo is what you need to defeat an enemy that's higher in FS and Luck/Skill than you. I like to develop one long combo for these times, where you can build the meter, execute the combo, and switch out to a throwaway character to bait and take out any enemy BBs that built. Rinse and repeat. If the enemy has a BB that can kill you, opt out of the long combo as these are often unsafe(red combo numbers mean a BB can punish you), tag out and bait the BBs first. SMs can't interrupt so don't worry about those.
Pick the right fighter for the right pizefight--for example, a BloodDrive Painwheel excells in any PF that applies a bleed to the opponent. So like the Medici Showdown(I believe).
Watch the enemy's BB meter--especially if the enemy has a BB3 you should keep an eye on this. Your hits charge the enemy with seemingly twice as much meter as the enemy's hits do. If the enemy has a BB available, play safe. Dashing out of your ground combo L3/4(depending) is a good one, since the AI will try to get you while you're unsafe, namely after you get to L5 or after they get a dash on you.
If you need meter fast--Sweeps, charge attacks, dashes, and most of all throws are the best way to gain this. Hold block in the middle of the throw animation in case they break it.
Knockback can be followed up on in a corner--Eliza and Fillia's throws apply a knockback that you can follow up on with Sekhmet or a Fillia-dash in the corner that could lead to another BB you can get on an enemy. Fillia's drill BB is a great example because you can get a hit or two in before using this as well.
Some SMs do more or the same damage as a full combo--while giving the enemy less BB meter. Painwheel's Pinion Dash and Parasoul's Napalm Shot(less than a full combo but close enough with that bleed) are great to use for this. Try Dash>L5>Launcher>SM for moves that won't hit on the ground like Napalm Shot.(Pinion Dash can be used after a full juggle though)
In Skullgirls console, there's something I don't know too much about where you can only do so many launchers or so until the enemy becomes invuln to it(I THINK it's one per reset). SGM has something similar where it seems like a character can't be hit when they're falling if they're below a certain height(as well as they can't be launched twice, they CAN be hit twice by a launcher, though). This makes corner combo's more lethal since you're still close to them when they're at that height, allowing a few more hits. Parasoul's umbrella-air-grab BB is able to be used when the enemy is falling, for example. It whiffs if the enemy has fallen below her waist-height though. Take this to the training room and see if you can combine more SMs, hits, and BBs into a falling combo.
That's all I can think of for now, but don't get too discouraged when you don't get to streak 15 or 20 >.>, as at that point any one of the enemy BBs will practically oneshot you, if they don't already. The odds are just stacked too high against anyone that doesn't have incredible luck/skill or a max character. I believe once you get your first max, or max team even, you can know how good you are by your streak.
You're right, I apply the technique you describe. I start the prize fights with characters that has low FS or level (until the 4th or 5th streak) to gain experience, then I use mid teams and finally when I reach the 9th or 10th streak I start to use my strongest characters.