For this achievement you must complete all levels on King Difficulty. The levels are as follows:
Lorien
Lorien Hollow
Mirkwood
Mirkwood Hollow
Thunderland
Poison Thunderland
Mirkwood Forest
Grakquarak
Blazing Grakquarak
Shadow Thunderland
Sealed Temple
To unlock King Difficulty you must complete the full story line campaign. After completing the Sealed Temple and returning to town, speak with Linus in Elvengard to officially start the quest.
It is possible to do this solo, but unless you are either a Fighter or a Slayer and have very good equipment and lots of resurrection stones, this will be extremely difficult unless in the company of at least one other person.
Being a Gunner myself, I was able to clear the first 3 levels solo, but came to a stand still on Mirkwood Hollow.
In the end, I made a run through with a partner (he was a Slayer). Together we conquered all levels in about 2 hours and I ended up using only about 10-15 Resurrection Stones.
Your best bet for finding partners would be to create a boosting session and join the boost list, the whole time I played this game I never encountered a random.
Best of luck traveling the King's Road!