What's the best way to get grass onto a dirt platform?

If you want to cheat - hack in a grass block, currently block ID 2.

Otherwise, one solution is to build a dirt walkway/staircase from the nearest patch of grass to your dirt. The thicker the walkway, the faster the grass will grow, as it has more room to spread.

One tip that I found useful is to place torches down (more the better), in order for the grass to grow at night - this speeds things up a lot :)

EDIT:

Check this thread for a helpful guide, in picture form! (note: some points on there may be out of date, as this is from an earlier build of Minecraft - the majority of it should be correct though)

With the silk touch enchantment on any tool, you can obtain a 'grass block' when digging instead of a dirt block. Place the grass in a patch of dirt and it will spread normally.

This is also helpful if you want to transplant Mycelium blocks.

Source: http://www.minecraftwiki.net/wiki/Enchanting

Now that there's pistons, just rig up a piston to push a grass block up to your platform!