The Burning Wheel supplement Path of Spite details the way to burn characters with Spite and to convert Spite back to Grief or to Hatred in play. But it doesn't present a way to convert Grief to Spite in play. So how does a character with Grief convert his Grief to Spite in play? Is it done during the normal trait vote?
There are two means: Trait vote and hearing the lullaby to sorrow.
Trait vote works as usual - roleplay your spite so that others vote it for your character. I'd determing the grief exponent as in character burning, with each significant item the character does not abandon or destroy counting as one bitter reminder. Another possibility would be to simply convert grief to spite at the same exponent (and shade).
Lullaby to sorrow, if successfully sung, gives the listener the choice to irrevocably turn their grief into spite. By rules as written, circles can be used to find dark elves, since every elven lifepath has a lead to the paths of spite. Or the player might take a relevant belief or instinct or actually talk to the GM and tell what they are thinking. Dark elves are an excellent failed circles result.
This answer is based on the old paths of spite PDF. Codex may have different rules.