I buy M tank and Eddie (for weapon energy purposes) before going in, get through the stage saving my T.Blow by using concrete shots getting through the first stage, and using four B.H.B. shots on the dragon (Pausing and shooting another one ensure quick work of the dragon). Use Magma Bazooka initially in boss fight to push back three of the spike balls then use Mega Buster for the rest of the fight.
Stage 2 - Use both weapon energy capsules to refill M. Bazooka and B.H.B. Primarily use Rush jet all the way to miniboss. Use charged up M.Bazooka on Clock Flower miniboss (A charged up shot will destroy the projectiles it flings at you) There are two options for avoiding the flower line, either run around or wait for it at the top and just jump over it, depends on where the flower likes to pop up.)
Use the B.H.B. on the water mines, in the following section I like to use the Rush Jet because the first jump with a spiked floor is my achilles heel and this is to make sure I make the jump. The following jump before the ladder I use a concrete shot as a stepping stone just in case I jump a bit early. J. Satellite is used til' the two giant enemies at the end, I use charged up M.Bazooka then uncharged shot or Plug Shots, depends on the amount of Weapon Energy I have left. For the first part of the boss I start with H. Chaser since the pod is out of reach and it also takes out the missles, once it lowers into jumping range, I use the L.Trident. Second section, make sure to have a FULL bar of T.Blow otherwise this part is a lot more tricky; Just unload and it'll be done in no time. Third part, just use L.Trident again, and B.H.B. if the lasers might hit the Blue Bomber.
Stage 3 - IMO, the easiest of the Wily Stages. Liberal use of weapons here until the M tank later in the stage. Before getting the M tank, I use the one previously bought, and use weapons sparingly. In the last section I memorized where the cannons are and inch forward so they barely show on the screen. This makes getting through them much more efficient, as the cannon will be aiming upwards which will allow you to hit them with the M.Buster, others just require quick fire. On the proto-man wannabes, I use the Plug Shot for the one above and the L. Trident for the last one since it'll hit him even when he has his shield up. This boss is extremely easy once you know the pattern. Use the B.H.B. and pause, then shoot again; If you're quick enough you can get three hits in one pass, just requiring two passes to do him in.
Final Stage - Gotta make each Concrete Shot count with the lasers to save as much as possible for the upcoming boss(2nd form of Wily). I take care of the bosses in a clock-wise fashion, I won't go into detail since there are FAQ's on the bosses already. I do everything but use the C. Shot on Galaxy man. I already talked about the second form earlier, so I'll just mention the last form. It's simple but not so much. When he's up in the corner and his bar is filling up, you can time it with either a Charged M.Bazooka or H.Chaser to get a hit in on him before he dissapears. This can useful because if you were just to use the Plug Shot, he will end up with 1 pixel of life left before the last shot so this would end the fight quicker. Sometimes he'll tend to appear right above you, and when he is about to shoot, use B.H.B. and he'll just sit there and quickly switch to the Plug Shot to get a hit or two in. Once you're out of B.H.B, use the M tank and you'll also have a full bar of T.Blow as well. If the fight drags on, you still have Eddie in hopes that he can provide some weapon energy capsules.
That right there is basically my thought process going through all the stages, it's just a matter of getting it right, and frustration creeps up when I'm so close and I get hit that I gotta blurb about it somewhere :/
in elephants almost ruined the game for me in the concrete level. But I came back and kept trying. I think they over-spiked in every level but the challenge is do-able. 
