Awesome that you're giving it a try, particularly at this point in the game's life!
Kahlan (from Champion Server) and Illenia (from Virtue server) were two of the first I heard about doing legit solo runs of this (without hacks, sticking LR, taking down towers and AVs from outside the warwalls, etc.) back on Live, I think both on Illusion Controllers, and I know with all the fun toys and improvements that came from I24/25 (or whatever it's being called now) that a number of people did it on the private server using things like /rad scrappers and brutes (rad makes sense for that TF as far as survivability) and even, as Corruption/FE said above, on Corruptors--although not without inspirations. Unless they've added a ton of other buff/debuff avenues in the last year or so or altered the TF in some way, I'd be surprised if a legit run could be done by most people on just about anything without at least some inspirations or the like, but who knows. It's much easier to solo a lot of these endgame TFs now than it was even at the beginning of Resurgence (or Beta or whatever it's called). It's cool to see people going for it and trying new ways and combos now that the game is busier again.
The main thing I've seen on the I24/I25 runs, aside from probably obvious elements like an optimally built character (IOs, ATOs, WOs, etc.) and incarnates, is liberal use of super inspirations at the right time. Keep plenty in the mail, primarily oranges and purples, and make sure you don't over-use them. The other element is just time. A lot of solo'ing TFs is just allowing yourself the time to experiment with different things, particularly the nuances of AV behavior, pulling AVs at need, dispersing damage or debuffs to neutralize enemies like those /Rad Aeon clones, and so forth. For a TW/Bio, I'd also recommend slotting for enough passive mitigation in your build, as you'll have plenty of damage from just the combo alone. That will make managing your super inspirations easier. Looking at the speed TF thread on this site, it looks like a number of people have soloed the TF on a TW/Bio, so you might shoot them some PMs and get some tips.
Good luck!