Let $V$ be a vector space over $\mathbb{C}$ endowed with a topology generated by a collection $\{p_\alpha\}_{\alpha\in A}$ of semi-norms.
I want to prove that this turns $V$ into a topological vector space. However, I didn't quite understood what does it mean for a function $$V\times V\to V\qquad\text{or}\qquad \mathbb{C}\times V\to V$$ to be continuous. Can I write this condition in terms of the semi-norms like I could do with a single norm?
To talk about continuity for a map $V\times V\to V$ or $\mathbb C \times V\to V$ you need a topology on the product. The product topology on the cartesian product is the (only) natural choice for this. It can be described in abstract terms (the coarsest topology on the product making both projections $(x,y)\mapsto x$ and $(x,y)\mapsto y$ continuous) or by specifying a fundamental system of seminorms. Here a natural choice for $V\times V$ is $P_\alpha(x,y)=\max\{p_\alpha(x),p_\alpha(y)\}$ (so that a ball with respect to $P_\alpha$ is the cartesian product of two balls w.r.t. $p_\alpha$) and $q_\alpha(t,x)=\max\{|t|,p_\alpha(x)\}$ for $\mathbb C\times V$. Now you can check continuity of the sum and the product using these seminorms. Since the sum $s:V\times V\to V$ is linear you only have to check continuity at zero, which follows from $p_\alpha(s(x,y)) = p_\alpha(x+y)\le p_\alpha(x)+p_\alpha(y) \le 2 P_\alpha(x,y)$. The product $\pi:\mathbb C\times V\to V$ is not linear (but only bilinear). I leave it to you to estimate $p_\alpha(\pi(t,x)-\pi(t_0,x_0)) \le c q_\alpha((t,x)-(t_0,x_0))$ for each $(t_0,x_0)$ and a constant $c$ depending on $(t_0,x_0)$ (write $tx-t_0x_0= (t-t_0)x+t_0(x-x_0)$).