These achievements require that you complete consecutive stages of Odyssey mode without losing a single life (crashing any planes). Start on Stage 1 to make this achievement easier, as you need to complete 9 in a row without losing a life, and the stages just get more difficult as you progress.
Completing stages is not related to score or landing a certain number of planes. It is only based on time, which is shown by the draining bar at the top of the screen. Knowing this, you can use it to your advantage to make earning these achievements (and for just completing the later stages) quite easy.
The technique involves exploiting the slow yellow ship that drops tiny yellow ships that remain stationary. The game spawns ships based on how many other ones you have on screen and whether you have already drawn lines to the runways for them. Basically what you want to do is anytime one of the large yellow ships spawns, draw a long path for it around the screen so it drops all five of the little yellow ships (each one only drops 5). Then organize the little ships in a cluster near the corners of the screen. Try not to make them too close to the corners because other planes can still spawn there. Pay particular attention to ships spawning near your clusters (the "!" indicators show up where ships are spawning, look for those). Make sure to move the spawning ships on paths away from your clusters. With the ships that do spawn, it is easiest just to move them around the screen killing time. If you land them, more planes will spawn. The picture below shows how it should look. One my first attempt, I got all the way from Stage 1 to 12 without losing a life and eventually just crashed some planes to end it out of boredom.
The Swift Swarm bot is also good to equip when playing Odyssey mode, especially when going for this achievement. Swarm ships (small green ones that look like bombs) are some of the most difficult to handle due to their sheer number and slow speed.