Consider the addition of two spins ##\vec{S}=\vec{s}_1+\vec{s}_2## and consider that ##\vec{s}_1## and ##\vec{s}_2## are realized in the irreps. with spins ##s_1## and ##s_2##, respectively, and we look for the reduction of ##\vec{S}## into irreducible parts of the induced "product representation".
This is achieved by noting that we have the product basis ##|s_1,s_2;\sigma_1,\sigma_2 \rangle##, which is the common basis of the compatible operators ##\vec{s}_1^2##, ##\vec{s}_2^2##, ##s_{1z}##, and ##s_{2z}## with (setting ##\hbar=1## for simplicity)
$$\vec{s}_1^2 |s_1,s_2;\sigma_1,\sigma_2 \rangle = s_1 (s_1+1) |s_1,s_2;\sigma_1,\sigma_2 \rangle,$$
$$\vec{s}_2^2 |s_1,s_2;\sigma_1,\sigma_2 \rangle = s_2 (s_2+1) |s_1,s_2;\sigma_1,\sigma_2 \rangle,$$
$$s_{13} |s_1,s_2;\sigma_1,\sigma_2 \rangle =\sigma_{13} |s_1,s_2;\sigma_1,\sigma_2 \rangle,$$
$$s_{23} |s_1,s_2;\sigma_1,\sigma_2 \rangle =\sigma_{23} |s_1,s_2;\sigma_1,\sigma_2 \rangle,$$
where ##s_1, s_2 \in \{0,1/2,1,3/2,\ldots \}## and ##\sigma_{13} \in \{-s_1,-s_1+1,\ldots s_1-1,s_1 \}##, ##\sigma_{23} \in \{-s_2,-s_2+1,\ldots,s_2-1,s_2 \}##.
Formally these basis vectors are given by the "product basis"
$$|s_1, s_2;\sigma_1,\sigma_2 \rangle = |s_1,\sigma_1 \rangle \otimes |s_2,\sigma_2 \rangle.$$
On the other hand we have ##\vec{S}^2##, ##S_3##, ##\vec{S}_1^2##, ##\vec{S}_2^2## as another compatible set of spin observables, i.e., we can build a common basis ##|S,s_1,s_2,\Sigma \rangle## with
$$\vec{S}^2 |S,s_1,s_2,\Sigma \rangle=S(S+1) |S,s_1,s_2,\Sigma \rangle,$$
$$\vec{s}_1^2 |S,s_1,s_2,\Sigma \rangle=s_1 (s_1+1) |S,s_1,s_2,\Sigma \rangle,$$
$$\vec{s}_2^2 |S,s_1,s_2,\Sigma \rangle=s_2 (s_2+1) |S,s_1,s_2,\Sigma \rangle,$$
$$S_3 |S,s_1,s_2,\Sigma \rangle = S |S,s_1,s_2,\Sigma \rangle.$$
Now the product basis is also an eigenbasis of ##S_3## and
$$S_3 |s_1,s_2;\sigma_1,\sigma_2 \rangle=(\sigma_1 + \sigma_2) |s_1,s_2;\sigma_1,\sigma_2 \rangle.$$
That means that the possible ##\Sigma \in \{(s_1+s_2),(s_1+s_2)-1,\ldots,-(s_1+s_2)+1,-(s_1+s_2) \}## and
$$\langle S,s_1,s_2,\Sigma|s_1,s_2;\sigma_1,\sigma_2 \rangle \propto \delta_{\Sigma,\sigma_1+\sigma_2}.$$
This implies that we must have ##S \geq s_1+s_2##, because otherwise there couldn't be the largest eigenvalue ##s_1+s_2## for ##S_3=s_{13}+s_{23}##. We can not have ##S>s_1+s_2##, because then via the ladder operator ##S_+## there'd be an eigenvector of ##S_3## with eigenvalue ##s_1+s_2+1##, but this doesn't exist within our product representation and so cannot exist in the other basis either.
Thus there's exactly one eigenvector ##|S=s_1+s_2,s_1,s_2,\Sigma=s_1+s_2 \rangle##, and we can choose it to be the product-basis vector ##|s_1,s_2;\sigma_1=s_1,\sigma_2=s_2 \rangle##. Thus we must have as one part of the new basis the complete basis to the irreducible representation with ##S=s_1+s_2##, which we can get from a repeated application of ##S_-=S_x-\mathrm{i} S_y## on this eigenvector with maximum ##\Sigma=s_1+s_2##, which finally stops with the eigenvector with ##\Sigma=-(s_1+s_2)##.
Now you consider the orthogonal complement of the subspace spanned by these ##2(s_1+s_2)+1## eigenvectors for ##S=s_1+s_2##. The largest eigenvalue for ##S_3## in this orthogonal complement can only be ##s_1+s_2-1##, and there's only one eigenvector to this eigenvalue (since there are two in the product basis, i.e., the ones where ##\sigma_1=s_1-1, \quad \sigma_2=s_2## and ##\sigma_1=s_1, \quad \sigma_2=s_2-1##, i.e., the eigenspace ##\text{Eig}(\Sigma,s_1+s_2-1)## is two-dimensional, and one basis vector we have constructed with ##S=s_1+s_2##. Now with the same argument as above for ##S## when restricting ##\vec{S}^2## to the orthonal complement to the ##S=s_1+s_2## irrep. we must have ##S=s_1+s_2-1##, and there's a unique (up to a factor) vector orthogonal to ##|S=s_1+s_2, s_1,s_2,\Sigma=s_1+s_2-1##, which we have to choose as the eigenvector of ##S=s_1+s_2-1,s_1,s_2,\Sigma=s_1+s_2-1##, and again we get all other eigenvectors for the irrep. with ##S=s_1+s_2-1## by acting repeatedly with ##S_-## to this vector.
Then we iterate the argument to the orthogonal complement of both the ##S=s_1+s_2## and ##S=s_1+s_2-1## subspaces etc. It becomes clear that the iteration stops with the lowest possible value ##S=|s_1-s_2|##. This you get by simple counting of the resulting dimensions. We know that the entire space is ##(2s_1+1)(2s_2+1)##-dimensional (spanned by the product basis) and the dimensions of the ##S## bases of the irred. subspaces just constructed iteratively add indeed up to (make ##s_1 \geq s_2## for simplicity)
$$[2(s_1+s_2)+1]+[2(s_1+s_2-1)+1]+[2 (s_1+s_2-2)+1]+\cdots + [2(s_1-s_2)+1]=(2s_2+1)2(s_1+s_2) - 2(1+2+\cdots 2s_2)+(2s_2+1)= 2 (2s_2+1)(s_1+s_2) -2 \frac{1}{2} (2s_1+1)(2s_2)+2s_2+1=(2s_1+1)(2s_2+1).$$
By explicitly doing this iterative scheme you can derive the Clebsch-Gordan coefficients
$$C(S,s_1,s_2,\Sigma|s_1,s_2;\sigma_1 \sigma_2)=\langle S,s_1,s_2,\Sigma|s_1,s_2;\sigma_1,\sigma_2 \rangle.$$
For your example we have
$$|S=1,s_1=1/2,s_2=1/2,\Sigma=1 \rangle=|s_1=1/2,s_1=1/2,\sigma_1=1/2,\sigma_1=1/2 \rangle,$$
$$|S=1,s_1=1/2,s_2=1/2,\Sigma=0 \rangle = \frac{1}{\sqrt{2}} (|s_1=1/2,s_2=1/2,\sigma_1=-1/2,\sigma_2=1/2 \rangle + |s_1=1/2,s_2=1/2,\sigma_1=1/2,\sigma_2=-1/2 \rangle),$$
$$S=1,s_1=1/2,s_2=1/2,\Sigma=-1 \rangle = |s_1=1/2,s_2=1/2,\sigma_1=-1/2,\sigma_2=-1/2 \rangle.$$
The remaining eigenvector with ##\Sigma=0## must be orthogonal to the above given with ##S=1## and ##\Sigma=0##, which uniquely (up to a phase factor) is
$$|S=0,s_1=1/2,s_2=1/2,\Sigma=0 \rangle=\frac{1}{\sqrt{2}} (|s_1=1/2,s_2=1/2,\sigma_1=-1/2,\sigma_2=1/2 \rangle -|s_1=1/2,s_2=1/2,\sigma_1=1/2,\sigma_2=-1/2 \rangle).$$