I have trouble finding a definition for the blow of of a point on the projective plane (or any projective space), and a projective plane curve. So first of all if you have a reference treating this I would appreciate. Hartshorne talks about affine varieties or blow-up coherent sheaves of ideals in general on a noetherian scheme but I didn't find anything on the concrete case of a point in a projective space. Gathman briefly says that on any algebraic variety we can consider an open covering of affine schemes, we can blow-up these and glue them. He also says we can define it to be the closure of the graph $(x,f_1(x),\dots,f_r(x))$ if $f_1,\dots,f_r$ are homogeneous polynomials defining our variety but it's not clear for me what this thing is supposed to be.
In his last comment on his answer Jérémy Blanc defined the blow-up of $\mathbb P^n$ at $[1:0:\dots:0]$ to be $$X=\{([x_0:\dots:x_n],[y_1:\dots:y_n])\in\mathbb P^n\times\mathbb P^{n-1}\mid x_iy_j=x_jy_i \forall i,j\in[1,n] \}$$ which I guess is a particular case of more general definition where we consider $[1:a_1:\dots:a_n]$.
Now consider the projective curve in $\mathbb P^2_{\mathbb C}$ defined by $f_{\lambda}=y^2z^3-x^5+5\lambda xz^4-4\lambda z^5$ with $\lambda\neq0,1$. It has a singular point at $[0:1:0]$ and so I want to blow it up at this point. I would guess that the total inverse image is $$\{([x:y:z],[t:u])\mid xu-tz=0,f_{\lambda }(x,y,z)=0\}.$$ Now if as in the affine case I consider the chart $U_t$ we would get $xu-y=0$ but replacing $y$ by $xu$ in $f_{\lambda}$ breaks homogeneity and so $[x:y:z]$ being a root of $f_{\lambda}$ doesn't make sense does it ?
Do we consider charts of $\mathbb P^1,\mathbb P^2$ silmunateously ? That is on the chart $U_x\times U_t$ for example we would get the that the total inverse image is $$\{([1:y:z],[1:u])\mid (y,z,u)\in V(z-u,y^2z^3-1+5\lambda z^4-4\lambda z^5)\}$$ which I'm not sure of. Is it the correct approach or am I off road ? And how to extract the strict transform from this ?
Finally if for example I consider the chart $\{y\neq 0\}$, I can blow-up as in the affine case. It seems to be enough (and simpler) to do this affine blow-up to characterize the blow up in $\mathbb P^2$, it it the case ?
In the chart $D(y)\subset \Bbb P^2$, your curve is cut out by $z^3-x^5+5\lambda xz^4-4\lambda z^5$. The blowup of $(0,0)\in \Bbb A^2\cong D(y)$ (= $[0:1:0]$ in $\Bbb P^2$) is covered by two charts, $\operatorname{Spec} k[x,z,t]/(tx-z)$ and $\operatorname{Spec} k[x,z,u]/(x-zu)$, with the maps to $\Bbb A^2=\operatorname{Spec} k[x,z]$ given by the spectrum of the natural maps $k[x,z]\to k[x,z,t]/(xt-z)$ and $k[x,z]\to k[x,z,u]/(x-zu)$.
To calculate the inverse image of your curve under these maps, you can substitute $z=xt$ in one chart and $x=zu$ in the other in to the equation $z^3-x^5+5\lambda xz^4-4\lambda z^5$ to get $$x^3t^3-x^5+5\lambda x^5t^4-4\lambda x^4t^4=x^3(t^3-x^2+5\lambda x^2t^4-4\lambda xt^4),$$ $$z^3-z^5u^5+5\lambda z^5u - 4\lambda z^5 = z^3(1-z^2u^5+5\lambda z^2u - 4\lambda z^2)$$ as equations of the total transform in each chart. To get the strict transform, remove the factor of $x^3$ or $z^3$ corresponding to the exceptional divisor so that you're left with the equations in the parentheses. Note that since $z=0,u=0$ is not a solution to the second equation of the strict transform, the strict transform is completely contained in the first chart.