Here is what I know/proved so far:
Let $c_0\subset\ell^\infty$ be the collection of all sequences that converge to zero. Prove that the dual space $c_0^*=\ell^1$.
$Proof$: Let $x\in c_0$ and let $y\in\ell^1$. We claim that $f_y(x)=\sum_{k=1}^\infty x_ky_k$ is a bounded linear functional. Clearly $f_y$ is bounded since
$$
|f_y(x)|=\left|\sum_{k=1}^\infty x_ky_k\right|\le\sum_{k=1}^\infty |x_k||y_k|\le||x_k||_\infty\sum_{k=1}^\infty |y_k|=||x_k||_\infty||y||_1.
$$
We can also easily see that $f_y$ is linear. Let $x,z\in c_0$ and $y\in\ell^1$ then
$$
f_y(x+z)=\sum_{k=1}^\infty (x_k+z_k)y_k=\sum_{k=1}^\infty (x_ky_k+z_ky_k)=\sum_{k=1}^\infty x_ky_k+\sum_{k=1}^\infty z_ky_k=f_y(x)+f_y(z)
$$
and for $\alpha\in\mathbb{R}$
$$
f_y(\alpha x)=\sum_{k=1}^\infty \alpha x_k=\alpha\sum_{k=1}^\infty x_k=\alpha f_y(x).
$$
Let $\varepsilon>0$ and since $y\in\ell^1$, we know that $\sum_{k=1}^\infty |y_k|$ converges. So there exists $N\in\mathbb{N}$ such that whenever $n>N$ we have
$$
\sum_{k=n}^\infty |y_k|<\varepsilon.
$$
Now define the following sequence $x=\{x_k\}_{k=1}^\infty$ as
$$
x_k=\begin{cases}
\operatorname{sgn}(y_k),&\,k\le N\\
0, & \,k>N
\end{cases}.
$$
Thus $x\in c_0$ and
$$
\begin{align}
\left|f_y(x)-||y||_1\right|&=\left|\sum_{k=1}^\infty x_ky_k-\sum_{k=1}^\infty |y_k|\right|\\
&=\left|\sum_{k=1}^N\operatorname{sgn}(y_k)y_k-\sum_{k=1}^\infty |y_k|\right|=\left|\sum_{k=1}^N |y_k|-\sum_{k=1}^\infty |y_k|\right|=\left|\sum_{k=N+1}^\infty |y_k|\right|<\varepsilon.
\end{align}
$$
So we conclude that
\begin{equation}
\ell^1\subseteq c_0^*.
\end{equation}
Observe that the above argument also establishes that $||f_y||_*=||y||_1$.
Now let $f$ be any linear functional on $c_0$ and let $\{e_k\}$ be the sequence with a 1 in the $k$-th position and zero elsewhere. Then for any $x\in c_0$ we have
$$
|f(x)|=\left|f\left(\sum_{k=1}^\infty e_kx_k\right)\right|=\left|\sum_{k=1}^\infty f(e_k)x_k\right|\le\sum_{k=1}^\infty |f(e_k)|\,|x_k|\le||x||_\infty\sum_{k=1}^\infty |f(e_k)|.
$$
Since $f$ is a bounded functional, we must have $\sum_{k=1}^\infty |f(e_k)|$ converging, otherwise $f(x)$ would be unbounded. Thus $\{f(e_k)\}_{k=1}^\infty\in c_0$ and we conclude that
\begin{equation}
c_0^*\subseteq\ell^1.
\end{equation}
Thus (1) and (2) tell us that
$$c_0^*=\ell^1.$$
QUESTIONS: If we take $c$ to be the collection of sequences that converge to some real number and $c^*$ to be it dual space, I know that $c^*=\ell^1$ as well, but I am not sure how to prove it. Is it enough to observe that if $x\in c$ and that $x\to x'$ then $x-x'\in c_0$, therefore they have the same dual space? I am a little fuzzy here, obviously.
Also, can someone clean up my align environment. I can't figure out how to make it compile properly. The code looks fine on my LaTeX implementation, but it doesn't work here.
Best Answer
The last argument where you say that $f(x)$ would be unbounded does not seem valid since you only have $f(x)\leq \Vert x\Vert_\infty\sum_{k=1}^\infty|f(e_k)|$. If $\sum_{k=1}^\infty |f(e_k)|=\infty$, you don't get any absurd. You could proceed as follows (assuming $f\neq 0$): For every $n$, the sequence $x^n=(\text{sgn}(f(e_1)),\ldots,\text{sgn}(f(e_n)),0\ldots)$ is in $c_0$ and has norm $\leq 1$, so $\sum_{i=1}^n|f(e_i)|=|f(x^n)|\leq\Vert f\Vert$. This show that $(f(e_1),f(e_2),\ldots)\in \ell^1$.
Now, about you next question: First, verify that $c=c_0\oplus\mathbb{R}$. Then $c^*=c_0^*\oplus\mathbb{R}^*=\ell^1\oplus\mathbb{R}=\ell^1$, where the last isomorphism is given by $((x_1,x_2,x_3,\ldots,),\lambda)\mapsto(\lambda,x_1,x_2,x_3\ldots)$.