The weight hierarchy of a $\mathrm{[}n\mathrm{,}k\mathrm{;}q\mathrm{]}$ linear code $C$ over $F_q$ is the sequence $\mathrm{(}{d}_{\mathrm{1}}\mathrm{,}\mathrm{\cdots }\mathrm{,}{d}_r\mathrm{,}\mathrm{\cdots }\mathrm{,}{d}_k\mathrm{)}$, where $d_r$ is the smallest support weight of an $r$-dimensional subcode of $C$. In this paper, by using the finite projective geometry method, we research a class of weight hierarchy of linear codes with dimension 5. We first find some new pre-conditions of this class. Then we divide its weight hierarchies into six subclasses, and research one subclass to determine nearly all the weight hierarchies of this subclass of weight hierarchies of linear codes with dimension 5.