You get more complete intersections if you relax your conditions a little. It is certainly more natural to ask for complete intersections in weighted projective space (WPS), not just straight (ordinary) projective space. A weighted projective space ${\mathbb P}^n[a_0,\ldots, a_n]$ with positive integer weights $a_0, \ldots, a_n$ is defined, as ordinary projective space, to be the quotient of ${\mathbb C}^{n+1}\setminus \{0\}$ by ${\mathbb C}^*$, except that the ${\mathbb C}^*$-action has weights $(a_0, \ldots, a_n)$ instead of the usual $(1, \ldots, 1)$. In the language of an ample divisor $L$ on a projective variety $X$, this corresponds to considering the whole graded ring $R=\oplus_k H^0(X, L^{\otimes k})$; it is finitely generated, by elements $x_0, \ldots x_n$, where $x_i\in H^0(X, L^{\otimes a_i})$. Then $R$ is a quotient of the free polynomial ring $S=k[X_0, \ldots X_n]$, dual to an embedding of $X$ into ${\mathbb P}^n[a_0,\ldots, a_n]$.

This language immediately gets you a few more cases. For example, suppose $X$ is a K3 surface of degree $2$. Then there is a two-to-one covering $\pi: X\to {\mathbb P}^2$. In the language above, we picked three degree one variables $x_0, x_1, x_2$ but the map you get is not quite an embedding yet, corresponding to the fact that the polarisation is ample but not very ample. It turns out that we are missing another variable $x_3$ of degree $3$, and $X$ embeds into the WPS ${\mathbb P}^3[1,1,1,3]$ as a sextic hypersurface, given (wlog) by an equation $x_3^2 + f_6(x_0, x_1, x_2)=0$. Here the sextic $f_6$ gives you the branch locus of the original map $\pi$.

More generally, you might be able to get your K3 surface as a complete intersection in some key variety, following terminology by Miles Reid. This says that there is some fixed variety $G\subset {\mathbb P}^n$ (often a Grassmannian) so that every K3 surface with a polarisation of degree $d$ is a complete intersection inside $G$. In this sense, K3 surfaces up to degree about 18 are complete intersections. This is very much related to Mukai's work reinterpreting the classification of Fano threefolds. One reference for all this is Brown-Altinok-Reid's http://xxx.lanl.gov/pdf/math/0202092.pdf, see in particular Theorem 2.5 for the statement I just made.

All of this is more related to algebra than to topology, in the sense that the structure of the graded ring $R$ determines whether or not it can be anywhere near a complete intersection in any key variety. One obstruction to a variety possibly being a complete intersection can already be read off the Hilbert or Poincare series $P_X(t)=\sum_{k\geq 0} h^0(L^{\otimes k})t^k$. By Hilbert's theorem, $P_X(t)$ is a rational function in $t$, with denominator (in the above language of generators of $R$) being equal to $\prod_{i=0}^n (1-t^{a_i})$. If $X$ is a complete intersection of hypersurfaces of degrees $d_1, \ldots, d_l$ in ${\mathbb P}^n[a_0,\ldots, a_n]$, then the numerator of $P_X(t)$ is $\prod_{j=1}^l (1-t^{d_i})$; this is a standard fact about regular sequences. Now the point is that for $(X,L)$ a polarised K3 surface of degree $d$, Riemann-Roch and Kodaira vanishing completely specify $P_X(t)$ in terms of $d$ (exercise!) and so you have your obstruction. More generally, the key varieties $G$ can be recognised (or at least guessed) from the numerator of $P_X(t)$; all this is explained in the above paper of Brown et al.

As a further set of examples, I wrote out the story of polarised elliptic curves before in this language here: Algebraic geometry examples