I will write some things here how I won Onslaught 5 times by now. First to win Onslaught in general, then for the Ziggs skin (and ward skin) and 5 man same champion. At the end I will give some tips to defeat Kayn specifically.
Onslaught in general:
If you do not want to have the same champs for all, then your team shoulf at least have: 1x Sona, 2x Yasuo, 1x Malphite and 1x Jinx/Ziggs. Your team should team up at all times and be running around Sona. Malphite is the one that should create combos to which the Yasuos can react. He is also the tankiest so he can wander of solo for a short period of time. The builds for this composition are pretty straight forward, just don't go on hit malphite, you will not make it.
Onslaught 5x same champ and <4 augments:
For this one I think Yassuo would be most suitable (I know of the other mission, but I haven't figured that one out yet). In this composition yiu should stick together as much as possible to clear the monsters as fast as possible. The augments you will need are: Tempest - Windwalker - Wanderer - Growing. (Leave growing out if you play for 3 augments). The building path should be: Berserkers - Phantom dancer - Infinity Edge - DEATHS DANCE - GUARDIANS ANGEL. The reason why I am caps locking the last two is because most people do not build these items. DEATHS DANCE: upon a GOOD ult you will be full health again regardless of how low you were. GUARDIANS ANGEL: this item resets after each level and can buy time for the revive spel/death timers.
After a good ult you will have approximately 10-15 tornados running to you. Do not waste those by standing in one place, but use them to kill monsters you did not hit in the first place. Use your ults EFFICIENTLY.
Apart from all the skill you might have, this also requires a lot of luck in terms of what Kayn does.
1. The main thing is to CC chain Kayn as long as possible and to keep attacking him while trying to dodge his abilities. Also you will need to stay grouped in order to survive for a prolonged time. This is essential as you don't have all day to kill him.
2. Kayn using "ult": run away from each other as fast as possible or it will kill you. If Kayn then spawns black holes --> lure Kayn to the other side of the map (!!)
3. Kayn uses dash: try to CC him, this will interupt his dash, and might even prevent him from continueing his dashing event for that time. If, however Kayn does continue to follow one player --> do not run all across the map, as this does only stall everything. Try to lure him back to the team as fast as possible.
4. Kayn's ball thingy: first kayn is charging up by absorbing souls. This is the time where you attack him non-stop. Use everything you have, except dashes (excluding malphite). When Kayn is very very very close to being done charging --> run away in different directions to make whatever Kayn fires dodgable. This moment can be postponed shortly when Yassuo(s) time the windwall correctly. (Ofcourse this does not apply for the things thay follow you on the ground, just try to kep walking in different directions all the time to dodge most).
I hope, by providing these small tips, you will come at least a bit closer to clearing Onslaught :). If there are any questions or additions, please post below.