Definition of a tensor? If a vector $\mathbf{F}$ is a linear function of another vector $\mathbf{A}$ such that $\mathbf{F}=C\mathbf{A}$ (the vectors are written as column matrices). Is the matrix $C$ a tensor? 
 A: You have to distinguish between a vector(object independent of a coordinate system) and a coordinate representation of that vector (triple of numbers identifying a vector in a specific coordinate system). The same is true for tensor (vector is a special case of a tensor).
One can either write equations where a symbol such as bold $\mathbf F$ means components of some vector, or where it means the vector itself. In practical matrix algebra it usually means a shorthand for the components, but in physics, the vector itself is usually meant. In order to talk about components, index notation is used, such as in the equation
$$
F_i = \sum_k C_{ik} A_k,
$$
where $F_i$ stands for three components of the vector $\mathbf F$ in a specific coordinate system. 
If there are two vectors which belong to the same vector space (in physics, if they are defined in the same inertial frame, like velocities of two particles $\mathbf v_1,\mathbf v_2$), one can introduce a tensor that relates them in the way you suggested.
There is infinity of different tensors that obey such condition. For example, we can write
$$
\mathbf v_2 = \left(\frac{1}{\mathbf v_1\cdot\mathbf v_1}\mathbf v_2\otimes \mathbf v_1 \right) \cdot \mathbf v_1
$$
so one possible tensor connecting the two vectors is
$$
\mathbf C = \frac{1}{\mathbf v_1\cdot\mathbf v_1}\mathbf v_2\otimes \mathbf v_1.
$$
This is a tensor because it is defined using only tensors ($\mathbf v_1,\mathbf v_2)$ and tensor operations (multiplying a tensor by a number, dyadic product $\otimes$ and dot product $\cdot$ of two vectors).
Alternatively, one may write this relation as
$$
v_{2,i} = \sum_m \left(\frac{1}{\sum_n v_{1,n}v_{1,n}}v_{2,i}v_{1,m}\right)v_{1,m}
$$
or
$$
v_{2,i} = \sum_m C_{im}v_{1,m}
$$
where $C_{im}$ is the matrix of tensor $\mathbf C$ in the chosen coordinate system:
$$
C_{im} = \frac{1}{\sum_n v_{1,n}v_{1,n}}v_{2,i}v_{1,m}.
$$
