Let's see what inferiority of one good in the two-good case implies. Look up Silberberg's "The Structure of Economics" (still one of the best undergraduate microeconomics textbooks ever written), ch. 10 for more details.
Utility maximization is described by (stars denote optimal levels)
$$U_A(A^*,B^*) - \lambda^*p_A \equiv 0$$
$$U_B(A^*,B^*) - \lambda^*p_B \equiv 0$$
$$y- p_AA^* - p_BB^* \equiv 0$$
and note the use of the identity symbol instead of simple equality -these relations always hold at the optimum. Then we can differentiate both sides and maintain the identity. Do that and solve the $3 \times 3$ system of equations to determine the various derivatives, and you will find that if good $A$ is inferior, $\frac {\partial A^*}{\partial y} <0$, then we must have that
$$p_AU^*_{BB}> p_BU^*_{AB}$$
If we are willing to accept $U_{BB} >0$, then the cross-partial $U_{AB}$ can be zero, and we can have a utility function as the one mentioned in @BKay 's answer.
But if we want to maintain $U_{BB} <0$, then it must be the case that $U_{AB}$, the cross-partial derivative of the utility function must also be strictly negative (and so not-zero). This in turn implies preferences that are not separable, additively or multiplicatively.
Perhaps you can consider something like
$$U(A,B) =\ln\left[aA^k + bB^h\right]$$
and all four parameters positive. For example, for values, $a=5, k=0.4, b=0.2, h=0.8$ the indifference map is

My conjecture is that for $0<h<1$ you may be able to have all the standard setup together with inferiority of $A$ (and for suitable values of prices and the other parameters of course). Find the first order conditions, substitute for $B$ in terms of $A$ in the budget constraint, and use the implicit function theorem to determine the conditions on the parameters required for $\frac {\partial A^*}{\partial y} <0$. And don't forget to check whether these conditions are compatible with the second-order conditions for utility maximization.
COMMENT October 7, 2015
Some comments in this answer appear to me to confound the issue of preference representation and preservation of preference ranking under monotonic transformations, with the "inferiority" property of a good. Preferences and their representation have nothing to do with the existence of a budget constraint. On the other hand, "inferiority" has everything to do with the existence of a budget constraint, and how it affects choices (not preferences) as it changes.
And monotonic transfomration do not leave everything "unchanged". Consider the utility function $V = A^k+ B^h$, and its monotonic transformation $U =\ln(A^k+ B^h)$. One can easily see that while $\frac {\partial^2 V}{\partial AB} = 0$, we have that $\frac {\partial^2 U}{\partial AB} \neq 0$. In other words, monotonic transformations may preserve the ranking of bundles, but this does not mean that they give the same relations between goods. And as I have written above, the property of "inferiority" depends on the signs and relative magnitudes of the second partial derivatives of the utility function used, signs and relative magnitudes that depend on the actual functional form used.