when you get double jump intentionly die you will respawn at start (if you skipped the first checkpoint) then go through the little gap and double jump up until you get to the peach section then double jump up to key maybe with so practise you can save about a second?
That is the route we all used to do, and you can find it in almost all runs.
If you are looking on the current WR run of mikerspd - he went upwards instead, saved at the computron, then went through all the enemies to get the green key from the top. After that he died intentionally and returned to the computron on top. This is a faster route to do, but it is dependent on how fast you get through the enemies. This can theoretically be done in clean motion (while killing only blue enemies and ignore the red ones), but very hard to do so.