I am asked to describe the defining representation of $\mathfrak{sp}_4$ in terms of highest weights, and then I am asked to repeat this process for the defining and adjoint representations of $\mathfrak{so}_5$
To start with, I noted that the subalgebra of diagonal matrices in $\mathfrak{sp}_4$ gave a Cartan Subalgebra $\mathfrak{t}$ that is spanned by the two diagonal $4\times 4$ matrices, $t_1 = diag(1,0,1,0)$ and $t_2 = diag(0,1,0,1)$.
Knowing this, suppose we take any vector $v = (a,b,c,d) \in \mathbb R^4$, then:
$t_1v = (a,0,c,0)$ and $t_2v = (0,b,0,d)$.
Hence, if $v$ were in the $\omega$ weight space of $V = \mathbb R^4$, then:
Either $\omega(t_1) = 0$ or $ b = d = 0$, and:
Either $\omega(t_2) = 0$ or $a = c = 0$.
Putting these two things together I concluded that $V = V_0$ is the $0$-weight space, and hence $V$ has highest weight $0$.
Second, noting that $\mathfrak{so}_5 \cong \mathfrak{sp}_4$, I argued that the defining representation of $\mathfrak{so}_5$ would behave in the same way, and thus would all be the $0$-weight space.
However, I feel like I must have made a mistake here somewhere because this doesn't seem right to me.
Additionally, I am unsure how to proceed to answer the part about the adjoint representation.
I think I will be taking the Cartan subalgebra $\mathfrak{t}$ of diagonal matrices in $\mathfrak{so}_5$, with some basis $t_1, t_2$.
Then I suppose I would take a general matrix $x \in \mathfrak{so}_5$ and compute $t_1x - xt_1 $ and $t_2x - xt_2$ to see if these are non-trivial scalar multiples of $x$. If so then $x$ can belong to some non-zero weight space of $V = \mathbb R^5$.
I wanted to ask though if there would be a better way to do questions of this kind.
The diagonal matrices you set up are not completely correct (since any matrix in $\mathfrak{sp}_4$ is tracefree). It should be $t_1=diag(1,0,-1,0)$ and $t_2=diag(0,1,0,-1)$. Now since weights are linear functionals, they are best expressed in terms of the dual basis to $\{t_1,t_2\}$ which usually is denoted by $\{e_1,e_2\}$. So $e_1$ extracts the first entry of a diagonal matrix and $e_2$ extracts the second of these entries. Now consider the standard basis $\{v_1,\dots,v_4\}$ of $\mathbb C^4$. By defintion $t_1v_1=v_1=e_1(t_1)v_1$ and $t_2v_1=0=e_2(t_1)v_1$, so $v_1$ is a weight vector of weight $e_1$. Similarly, the other $v_i$ are weight vectors with weights $e_2$, $-e_1$ and $-e_2$, respectively. The standard ordering of weights is that $e_1>e_2>0$, so the highest weight is $e_1$. For the adjoint representation you have to work similarly using an appropriate basis of $\mathfrak{sp}_4$ consisting of matrices with only one or two non-zero entries. Just give it a try. For $\mathfrak{so}_5$ you first need a Cartan subalgebra, then you can analyze the representations similarly, showing the as stated in the comment of @David_Hill that the standard representation of $\mathfrak{sp}_4$ corresponds to the adjoint representation of $\mathfrak{so}_5$ and vice versa.