Here's the way I do the lava beam section:
1. Fill up all your weapons and life energy to maximum using the Jewel Satellite on those floating guys on the screen below the lava beams.
2. Climb the ladder to the first lava beam room. Move beneath the ladder that goes to the next room up. Call Rush Coil (or, if you're Proto Man, the Proto Coil) and when the lava is clear, hop on and grab the ladder.
3. Climb to the next room up. As soon as you are standing on top of the ladder, face right and climb back down so you don't get killed by the soon-incoming lava. You want to make sure you're facing right for this next bit. Leave the screen by climbing down and then climb back up again. Here's where you need to be quick—a minor delay will probably mean death.
4. Climb to the top of the ladder (you should now be facing right) and call your Coil again. Hop on immediately and grab the left-most ladder. Climb up enough that you'll be clear of the top lava beam. If you want to pick up an Energy Tank, read 4a. Otherwise, skip to 5.
4a. Climb to the top of the ladder on the next screen up. As soon as you're standing on top of it, jump up to the next ladder, which is directly above your head. Climb up it and carefully jump to the next ladder, to the left. If you want to take an extra precaution, shoot a Concrete Shot—not at the lava, but at the opposite wall, to be used as a platform to make reaching the ladder a little easier. Climb the ladder, shoot the blocks with a Laser Trident, claim your Energy Tank, and then climb back down. Jump down to the ladder you used at the beginning of this step to climb onto this screen in the first place, and climb down to the screen below.
5. While still hanging from this ladder, wait for the topmost lava beam to be fully extended across the screen. Shoot it with a Concrete Shot, then jump down onto it and quickly make your way to the rightmost ladder. Climb up it.
6. Climb to the top of the ladder on the next screen up. As soon as you're standing on top of it, jump up to the next ladder, which is directly above your head. Climb up it to the next screen, and to safety.
Hopefully this was clear. There is that one quick little step there where you have to be fast, but other than that it's all straightforward and fairly easy.