I realise this question was asked a long time ago, so my answer here may be tumbling into the void unread, but I'll answer it anyway, and hope that any interested parties reading this question will benefit from an answer.
The definition of cyclic cohomology that you described here is not quite correct. There are a couple technical points that are missing so I'll give a run down of the definition and hopefully that should answer the question.
The definition that it seems you are referring to is not typically used for $k$-algebras where $k$ is an arbitrary field. You will normally see it defined in this way for $\mathbb{C}$-algebras. So with this in mind, let $A$ be a $\mathbb{C}$-algebra and consider $\text{Hom}_{\mathbb{C}}(A^{\otimes n+1}, \mathbb{C})$. A map $f\in\text{Hom}_{\mathbb{C}}(A^{\otimes n+1}, \mathbb{C})$ satisfying
$$f(a_0\otimes \dots\otimes a_n) = (-1)^nf(a_n\otimes a_0\otimes \dots\otimes a_{n-1})$$
is called a cyclic cochain. Then we define $C^n_\lambda(A)$ to be the cyclic cochains in $\text{Hom}_{\mathbb{C}}(A^{\otimes n+1}, \mathbb{C})$. It can be shown that the image $b(f)$ of a cyclic cochain $f$ under the map
\begin{align}
b(f)(a_0\otimes\dots\otimes a_{n+1}) &= \sum_{i=0}^n(-1)^if(a_0\otimes\dots\otimes a_ia_{i+1}\otimes\dots\otimes a_{n+1})\\
& \quad +(-1)^{n+1} f(a_{n+1}a_0\otimes\dots\otimes a_n)
\end{align}
is again a cyclic cochain, and that $b^2=0$. The cyclic cohomology $HC^{\ast}_\lambda(A)$ is then defined as the cohomology of the cochain complex $(C^{\ast}_\lambda(A), b)$.
The way you relate the cochain complex $(C^{\ast}_\lambda(A), b)$ to the Hochschild cochain complex is in the following way. The Hochschild cochain complex of $A$ with coefficients in an arbitrary $A$-bimodule $M$ is given by
$$M \xrightarrow{d} \text{Hom}_{\mathbb{C}}(A, M) \xrightarrow{d} \text{Hom}_{\mathbb{C}}(A^{\otimes 2}, M) \xrightarrow{d} \cdots$$
where $d$ is the Hochschild coboundary map
\begin{eqnarray}
d(f)(a_1\otimes \cdots \otimes a_{n+1}) &=& a_1f(a_2\otimes \cdots \otimes a_{n+1}) \\
&&+ \sum_{i=1}^{n}(-1)^{i}f(a_1\otimes \cdots \otimes a_ia_{i+1}\otimes \cdots \otimes a_{n+1}) \\
&& + (-1)^{n+1}f(a_1\otimes \cdots \otimes a_n)a_{n+1}
\end{eqnarray}
Defining $C^n(A, M):=\text{Hom}_{\mathbb{C}}(A^{\otimes n}, M)$ then the Hochschild cochain complex is the cochain complex $(C^{\ast}(A, M), d)$. Now, consider the particular case where $M$ is the dual space $A^{\ast}$ of $A$, that is, $M=A^{\ast}=\text{Hom}_{\mathbb{C}}(A, \mathbb{C})$. Then, writing $C^n(A):=C^n(A, A^{\ast})$, we have
$$C^n(A) = \text{Hom}_{\mathbb{C}}(A^{\otimes n}, \text{Hom}_{\mathbb{C}}(A, \mathbb{C}))$$
but from the basic properties of modules we know that for $R$-modules $M, N$ and $P$ where $R$ is a commutative ring
$$\text{Hom}_R(M, \text{Hom}_R(N, P))\simeq \text{Hom}_R(M\otimes_R N, P)$$
and therefore
$$C^n(A)\simeq \text{Hom}_{\mathbb{C}}(A^{\otimes n+1}, \mathbb{C})$$
and thus $C^n_\lambda(A)\subset C^n(A)$. We can also show that under this identification the Hochschild coboundary map $d$ is equivalent to $b$.
Therefore $(C^{\ast}_\lambda(A), b)$ is a subcomplex of the Hochschild cochain complex $(C^{\ast}(A), d)$ of $A$ with coefficients in $A^\ast$. This inclusion of complexes then induces a map $I:HC^n_\lambda(A)\to HH^n(A)$ for each $n\ge 0$, where $HH^n(A):=HH^n(A, A^\ast)$ is the Hochschild cohomology of $A$ with coefficients in $A^\ast$. Note that $I$ may not be injective. You can see this by comparing $HC^\ast_\lambda(\mathbb{C})$ and $HH^\ast(\mathbb{C})$. For $n>0$ we have $HH^n(\mathbb{C})=0$ and for $n\ge 0$ we have $HC^{2n}_\lambda(\mathbb{C})=\mathbb{C}$. Therefore $I:HC^{2n}_\lambda(\mathbb{C})\to HH^{2n}(\mathbb{C})$ is not injective for $n>0$.
As a historical note, the definition I've given here is the definition that was first given by Alain Connes, who introduced cyclic cohomology as an analogue to de Rham homology for noncommutative spaces and is the definition that you will come across in a lot of texts and notes on the subject, such as Connes' own writings, and others like Khalkhali or Brodzki. No doubt there are more that I've forgotten or not seen.
There is another definition that is used for associative $k$-algebras where $k$ is an arbitrary field. This is the one you find in Loday and Weibel, and elsewhere. To be absolutely precise, it's for unital associative $R$-algebras, where $R$ is a commutative ring, but we'll stick with a field since it's what we've been working with so far. The motivation for it is a little more technical so I'll just state it without much context. Although I will say that it arises out of a cyclic group action on the Hochschild complex.
Let $A$ be a unital associative $k$-algebra where $k$ is a field. Then let $t$ be the $\mathbb{Z}/(n+1)\mathbb{Z}$ action on the generators of $C^n(A)=\text{Hom}_k(A^{\otimes n+1}, k)$ given by
$$ tf(a_0\otimes\dots\otimes a_n) = (-1)^nf(a_n\otimes a_0\otimes\dots\otimes a_{n-1}) $$
which we then extend by linearity to $C^n(A)$. Then let $N:C^n(A)\to C^n(A)$ be defined by $N = \sum_{i=0}^nt^i$, and $b^\prime$ the Hochschild coboundary map with the last term removed, that is
$$b^\prime(f)(a_0\otimes\dots\otimes a_{n+1}) = \sum_{i=0}^n(-1)^if(a_0\otimes\dots\otimes a_ia_{i+1}\otimes\dots\otimes a_{n+1})$$
then these maps give rise to a first quadrant bicomplex
$\require{AMScd}$
\begin{CD}
\cdots @. \cdots @. \cdots @. \cdots \\
@AAA @AAA @AAA @AAA \\
C^2(A) @>1-t>> C^2(A) @>N>> C^2(A) @>1-t>> C^2(A) @>N>> \dots \\
@AbAA @A{b^\prime}AA @AbAA @A{b^\prime}AA \\
C^1(A) @>1-t>> C^1(A) @>N>> C^1(A) @>1-t>> C^1(A) @>N>> \cdots \\
@AbAA @A{b^\prime}AA @AbAA @A{b^\prime}AA\\
C^0(A) @>1-t>> C^0(A) @>N>> C^0(A) @>1-t>> C^0(A) @>N>> \cdots
\end{CD}
which we denote by $CC(A)$. Then the cyclic cohomology $HC^\ast(A)$ of $A$ is defined as the cohomology of the total complex of $CC(A)$, that is
$$HC^n(A) = H^n\text{Tot}(CC(A))$$
it can then be shown that when $\mathbb{Q}\subset k$ (or equivalently, when $k$ has characteristic $0$) then the inclusion $(C^{\ast}_\lambda(A), b) \hookrightarrow (C^\ast(A), d)$ induces an isomorphism $HC^n_\lambda(A)\to HC^n(A)$ for all $n\ge 0$. So that if $A$ is a $\mathbb{C}$-algebra then both of these definitions of cyclic cohomology are equivalent. So this definition makes calculations possible for a much wider class of algebras, although in general the calculations are harder to compute.
This may be past the scope of the question, but if you're interested in cyclic cohomology it's certainly worth knowing.
