What's the best way to get grass onto a dirt platform?
Solution 1:
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
Solution 2:
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 (note: some points on there may be out of date, as this is from an earlier build of Minecraft - the majority of it should be correct though)
Solution 3:
Now that there's pistons, just rig up a piston to push a grass block up to your platform!