What the achievement asks
Reach Renown rank 10 with the Flame's Radiance. Ten points, account-wide.
What the Flame's Radiance is
The Flame's Radiance is the faction attached to the Siren Isle and its storm content. Its renown comes from the isle itself: the weekly quests, the excavations, the rares and the events that run there.
There is no separate grind to plan. Ten ranks arrives from doing the isle's own content, and the isle's own content is almost entirely other achievements in the same meta.
The route is the weeklies
The seventeen weekly quests of Siren-ity Now! are the largest and steadiest source. They rotate across three invasion types with a different set each week, so doing every quest available on every reset is both the fastest renown and the only way to finish that achievement.
Alongside them, the excavations of Excavation Projects, the sixteen rares of Clean Up on Isle Siren and the caches of Treasures of the Storm all pay in.
Start with A Song of Secrets, the isle's questline, since it opens everything else.
Expect three or four weeks
This is a calendar achievement rather than an effort one. The weekly quests are capped by the reset and the invasion rotation takes three weeks to show you all of them, so renown 10 lands somewhere around the third or fourth week of visiting the isle regularly.
That makes it a poor thing to leave until last. Anyone chasing the War Within meta should be doing a Siren Isle lap every reset from the beginning, and this achievement then completes itself while the others do.
Where it fits
A piece of Worldsoul-Searching, the War Within meta, and a branch of its own rather than a step inside one.