Go-to pressure strings are staggers of b1212 (b12 being the best), s1 and 11, d4, f3, and of course the almighty d1 check.
His pressure is a lot like jax's with the constant jabs and button pressing, eventually you want to command grab the opponent or condition them to think you're going to grab, thereby opening them up. Don't play him like a 50/50 machine because his overhead is super slow and reactable, but you can go for it occasionally (it's very safe and confirmable).
One of the best ways for lackey to open up the opponent is committing to something slightly negative (f2, 111, b121 or b1212, 2f4 etc) and then armoring through your opponent's button press. EX Torryuken can lead to a 45-46% combo every time, so the damage adds up. This gets the opponent scared to take their turn back, so you can get them conditioned to let you press buttons for days.
Here's some tips for your offense:
D1 canceled into command grab for a tick throw is way too risky for the small reward it gets. It's better to wait and see if the d1 hits and then go for a command grab after.
B12 on block leaves you with tons of options. It spaces you out perfectly to go for another b12, or raw command grab, or jump in 2, or just a simple d1 check.
Don't be afraid to mash d3, and occasionally d1. I'm serious, torr's d3 is stupidly good on block.
You can't get away with this BS against everyone, but EX torryuken into another EX torryuken on block is a thing.
Getting around zoning/keepaway tactics is what Lackey sucks at. His charge special will eat one projectile if timed right but 99% of the time you'll just get hit out of it by the next fireball. EX will get you in, but also get you full combo punished by pretty much anyone (aka not worth it). EX roll travels the screen pretty quickly but your timing has to be perfect because it loses its armor about halfway through the animation, and it doesn't low profile shit. This, along with his horrible walk and run speed means that you'll be spending 90% of the match getting zoned out (if your opponent plays the matchup right). You can even get zoned by non-zoners (fucking bo rai cho keeps lackey out free) and you will basically just be going through an obstacle course with shit mobility if you want to ever touch your opponent. That means when you get in, you have to get that damage and make it count, and when you get them cornered you have to do everything you can to keep them there.
That's my rundown of the variation, hope you enjoy playing lackey and experiencing the struggle. It's fun. I have a thread somewhere on the f/t forum with a video of bnb's and setups so you can get those there.