Approaching is a major weakness for Luigi since he lacks range. He does have some nice approaching tools, though.
1. Fireball: Use it on the ground or even in the air. Hell, you can spam it if you like! It will help Luigi get in on an opponent who shielded a Fireball or got hit by one. By that point, Luigi can rush in and grab. Don't do this too often as you will be pretty predictable.
2. Short Hop Fair / Dair / Bair: Luigi has powerful aerials, use them to your advantage! The attacks don't have the greatest range, but they will help Luigi get in and it is a good alternative for him. Sometimes opponents will shield the aerials; fast fall and THEN grab or follow up with yet another move.
3. Luigi Cyclone: This is a multi-hit move that can create nasty setups and combos. You can use the Luigi Cyclone to follow up with another move or aerial. You have to be careful that this isn't always the safest approaching tool for Luigi as it can be easily blocked as well; the ending animation for the Luigi Cyclone will leave him wide open and you will get punished. I would alternate this one in between his other approaches.
You can also use the Green Missile, but I am not a fan of that one since you will get a good hit or look silly for even using it. I wouldn't use the dash attack as a primary way to approach, but I would occasionally mix it up if my opponent rolls a bit too much.