Flint & Steel
In order to gain Obsidian you need to find a place underground where lava mixes with water, and then mine said spot. Said block looks like this
Although in order to mine Obsidian you need to mine it with a Diamond pickaxe. To gain a diamond pickaxe you need to mine 3 diamond rocks with an iron pickaxe or higher and then craft them using the crafting machine. Diamond rocks looks like this
Gaining Flint & Steel
Flint & Steel is created by using flint and an iron ingot (Derived from Iron ore) in a crafting machine.
Building the Portal
With your Obsidian blocks build the basic portal shape, 4 blocks wide and 5 blocks tall. Then use your flint & Steel on top of one of the Obsidian blocks on the bottom of the portal. The empty space inside the portal will turn purple and voila, achievement unlocked.
Friend who has a nether portal already constructed
Join your friends world and meet him at the portal, if the portal is alight (Purple glow in the empty space) mine away an obsidian block and when the block is replaced the portal will be unlit.
Grab your flint & steel and simply re-light the portal. This will net you the achievement.
Update #1 - Credit to Eskimoletdinov
Instead of finding a unique mine in which both water and lava co-exist it is possible to bring along buckets of water, and use them on the lava to make Obsidian. Be careful though as you may slip into the lava and die!
To create buckets you need to have iron ore, then smelt it into an iron ingot, and then create a bucket with it in the crafting table.