Let $E$ be a vector space over a field $F$ and $u$ a linear map $E \rightarrow E$ which is nilpotent, ie, $u^p = 0$. I want to show that there exists a decomposition $E = F_1 \oplus ... \oplus F_p$.
To do this, I know that there exists an ascending chain of subsets $\ker{u} \subset \ker{u^2} \subset ... \subset \ker{u^p} = E$. This suggests we can construct the direct sum by taking "complements". Let's try and construct the $F_i$ such that $\ker{u^j} = \oplus_{i=1}^j F_i$. We need to take the correct "complements" in the ascending chain of subsets. The first thing I think of is to begin with a basis of $\ker{u}$, eg, set it to be $B_1 = \{e_1, ..., e_{\dim{\ker{u}}}\}$. Then the basis of $\ker{u^2}$ can be extended from $B_1$, ie it's $B_2 = \{e_1, ..., e_{\dim{\ker{u}}}, e_{\dim{\ker{u}} + 1}, ..., e_{\dim{\ker{u^2}}}\}$.
Now we can construct the $F_i$. Let $F_1$ be a vector space with basis $B_1$ and $F_2$ have basis $\{e_{\dim{\ker{u}} + 1}, ..., e_{\dim{\ker{u^2}}}\}\}$ Then $\ker{u^2} = F_1 \oplus F_2$. We can construct more $F_i$ similarly. We get $\ker{u^j} = \oplus_{i=1}^j F_j$ and therefore we end up with a decomposition $E = \oplus_{i=1}^p F_p$, as required.
Does that sound right?