You must start your quest for a trophy on the first day of a month. That is when all the high scores are cleared. You should be on the first quest. The key to earning a trophy is speed. You need to minimize the elapsed time between when you accept a quest and complete the quest. You want to earn at least 1980 points per quest (my best was 1990, my average was 1984). The second most important point is that you also must be able to complete 2 quests per day. You can accept a quest every 12 hours.

Keys to speed:

Have plenty of NP available.

Try to get ready a few minutes prior to the when your next quest becomes available.

Have 2 browser windows open. One window must display the Illusen's/Jhudora's quest window and one with a Shop Wizard. Set the Shop Wizard 'Search Items' to "identical to my phrase". Accept the quest. Copy the name of the item needed (Ctrl†C or Ctrl†Insert). Paste (Ctrl†V or Shift†Insert) the name of the item into the Shop Wizard 'What are you looking for?' Box. Hit Enter or Click the Search Shops button. Click on the first item in the list and buy it from the shop. As soon as you buy it then go back to Illusen's/Jhudora's quest window and click to complete the quest. Do not wait for the shop screen to refresh.

You can eliminate most of the things that can slow you down by following the Keys to speed above. But you are at the mercy of Neopets and your Internet Access provider. Also a shop with a lot of graphics can slow you down. It might be quicker to choose a shop with only 1 item compared to more then 1. My theory is that someone who focuses on shopping and has more items in their shop tends to have fancier shops (more graphics) and lots of items. Both may slow you down.

If you start out in the top 50 then you have a good chance of getting a high score if you complete 2 quests per day. All it takes is for someone above you to only complete one quest in a day and they are history. Some people might stop accepting quests once they earn the trophy. Preferably you have access to the computer at 12 hour intervals (i.e. 8am and 8 pm). If you are late on one quest then all the rest of your quests will be at a later time.

I have not done this but I beleive some people have managed to get in an extra quest on the first day. They start their first quest shortly after all the high scores are cleared (Midnight - 4:00 am NST on the first of the month) and if the high scores are next checked greater then 24 hours from that time then someone could get three quests in and have a leg up on every one else.

Good luck!