The answer to your first question is roughly yes, although in general a slice category may not be a subcategory since the image of an object in the source can admit many maps to the target, and these all give distinct (possibly non-isomorphic) objects in the slice. Of course, you can still define the pullback along the forgetful functor.
As for your second question, this is just the fact that the "tensoring up" functor $\mathbb{C}[H] \otimes_{\mathbb{C}[G]} - :\mbox{Rep}(G) \rightarrow \mbox{Rep}(H)$ is adjoint to the forgetful (or "restriction of structure") functor $U:\mbox{Rep}(H) \rightarrow \mbox{Rep}(G)$.
To actually compute this colimit, let's write $\mathcal{G}$ and $\mathcal{H}$ for the categories, $*_G$ and $*_H$ for their unique objects, and identify the morphisms with the sets $G$ and $H$. Then the objects of $(f\downarrow *_H)$ are given by pairs $(*_G,h:*_H \rightarrow *_H)$ (for all $h\in H$), and the morphisms of $(f\downarrow *_H)$ are given by $ \mbox{Hom}_{(f\downarrow *_H)}((*_G,h_1:*_H \rightarrow *_H),(*_G,h_2:*_H \rightarrow *_H) = \{g\in G : f(g)=h_2^{-1}h_1\} $ (using the standard convention for composition).
Now we can explicitly describe the diagram $(f \downarrow *_H) \xrightarrow{U} \mathcal{G} \xrightarrow{F} \mbox{Vect}$: it takes every object to $V$, and to the morphism $f(g)=h_2^{-1}h_1$ it associates the linear homomorphism (actually isomorphism) $F(g)$ from the copy of $V$ over $h_1$ to the copy of $V$ over $h_2$.
Colimits are built as follows: given a diagram $D:\mathcal{I} \rightarrow \mathcal{C}$, $ \mbox{colim}(D) \cong \mbox{coeq} \left( \coprod_{(i_0 \rightarrow i_1)\in \mbox{mor}(\mathcal{I})} D(i_0) \rightrightarrows \coprod_{j \in \mbox{ob}(\mathcal{I})} D(j) \right),$ where one map takes the $D(i_0)$ over $(i_0 \rightarrow i_1)$ to $D(i_1)$ over $j=i_1$ via $D(i_0 \rightarrow i_1)$, and the other map takes $D(i_0)$ over $(i_0 \rightarrow i_1)$ to $D(i_0)$ over $j=i_0$ via the identity map. Of course, in $\mbox{Vect}$ all this means is that you take a big direct sum of all the vector spaces in the diagram and then take the quotient where we identify $v\in D(i_0)$ with $(D(i_0 \rightarrow i_1))(v) \in D(i_1)$ (for all $(i_0 \rightarrow i_1) \in \mbox{mor}(\mathcal{I})$).
So for our situation, we start with the direct sum of a bunch of copies of $V$ indexed by $H$, i.e. $\bigoplus_H V \cong \mathbb{C}[H] \otimes V$. (Categorically, one would say that $\mbox{Vect}$ is tensored (or copowered) over $\mbox{Set}$, which essentially means that we have a reasonable and (bi)functorial notion of "taking coproducts of an object indexed by a set".) Then, if we have a vector $v\in V$ considered as sitting over $h_1$ -- that is, the element $h_1 \otimes v$ -- and $f(g)=h_2^{-1}h_1$ is a morphism from $h_1$ to $h_2$, then we make the identification $h_1 \otimes v \sim h_2 \otimes g \cdot v$. But notice that this is just the usual definition of trading off the right action of $\mathbb{C}[G]$ on $\mathbb{C}[H]$ via $f$. (To be precise, this action comes from the functoriality and the product-preservation of the tensored structure I mentioned above.) So finally, $\mbox{colim}_{(f \downarrow *_H)} F \circ U \cong \mathbb{C}[H] \otimes_{\mathbb{C}[G]} V$.