I used to do solo l5s back then when the agents were giving out the missions in highsec (it was some bug IIRC, and was fixed a long time ago). I only took the anti-pirate and anti-drone missions and completely ignored all vs minmatar and gallente (my standing was bad enough with them anyway). my personal favourite mission was gurista honor. I did all of those missions in a full passive drake solo first, and it was possible to tank them (the only one which was impossible to solo in a drake was part 3 of the rogue drone series of 3 missions). Of course it took AGES because my DPS sucked. Later I changed to a Nighthawk and a passive Tengu, which was a massive improvement, and even more so the passive Rattlesnake I used in the end. But nowadays I would prefer a pack of Tengus for the extra mobility (missions will be in low sec, so be prepared to get visitors).
You should make yourself familiar with the resistance profiles you need and go in with a maximum tank first approach, with more experience you will see where you can swap tank for damage to get the mission done faster. The Tengu has a good native thermal resistance, and deals best kinetic damage, so missions vs Gurista or Serpentis should be best (AFAIK there is only 1 anti-Gurista mission in the l5 mission pool though). When fighting Angels also use Scourge missiles. Good luck