Technically, this trophy's description should read 'complete 10 levels on one save file'.
You do not have to complete 10 levels in a row without dying. If in each run, you can only complete one full level then you can use 'stage select' to start again from the highest level you have unlocked.
Keep playing until you unlock stage 10 and then beat it for this trophy. Stage 10 is the last level of the game; all stages beyond that are looped again from stage 1.