I created a video achievement guide for the four (in the video I say three so please disregard that) chain reaction achievements. If you focus on making a 15 second chain reaction you will get the other three achievements along the way.
Video Achievement Guide Link:
https://onedrive.live.com/redir?resid=1883FA8DF9FE1EE1%21157...Using my video guide you will earn the following achievements:
In addition to the video, I will also offer a text alternative but I recommend watching the video too of course.
First off, I did these achievements in the 60 second mode but infinity mode would work as well so the mode isn't that important. As you play the game and collect coil powerups, you will be progressing towards new unlocks (ie: missile powerups, new skins/backgrounds, etc). It won't take long to unlock them all but its not required for these achievements, though it does make it easier if you have missiles unlocked.
Powerups such as missiles will randomly show up sometimes and they will be one of the yellow powerups. Once you have at least one missile, try to round up several big enemies together. Don't shoot them right away. Instead, try and make them follow you and go in a pack. When a few of them are surrounding you, press RT to use a missile.
If you killed enough of them (your chain went past x50), you will now see the words "chain reaction" under your ship. This will slow the game down giving you more time to keep the chain reaction going. As long as you hit another enemy within about 2 seconds, the chain reaction will continue. Your goal is to get the chain reaction to last 15 seconds (EDIT: it's actually longer than 15 real seconds since the game timer at the top is slowed down. Just try to keep the chain going as long as possible just to be sure).
By far the best and easiest way to keep your chain going is to find and shoot a red enemy. These enemies are semi-invincible, which is good because as long as you shoot a red enemy, the chain will continue! Now, what I mean by semi-invincible is that your main gunfire will not kill them however other explosions from other nearby enemies, bombs, missiles or other powerups will kill kill them. If that happens, you still have a small window of time to find and shoot another enemy (preferably another red one) to keep the chain reaction going. If you lasted 15 seconds or more, the achievement will pop after your game ends.