Bardsley and Richardson (Etale slices for algebraic transformation groups in characteristic p.
Proc. London Math. Soc. (3) 51 (1985), no. 2, 295–317) give a construction which seems to do what you want. It gives less than a "Cayley map" as in Borovoi's answer.

Let $G$ be connected and semisimple over a field of char. 0.
Bardsley and Richardson construct a mapping $G \to \operatorname{Lie}(G)$ with
nice properties (which I'll indicate below).

Note that we may as well suppose that $G$ is of adjoint type -- indeed,
if the problem is solved already for the adjoint group $G_{\operatorname{ad}}$,
just take the composite
$$G \to G_{\operatorname{ad}} \to \operatorname{Lie}(G).$$

Now, since the characteristic of $k$ is zero and $G$ is of adjoint type, the adjoint representation
$V = \operatorname{Lie}(G)$ is a faithful representation of $G$ for which the
trace form
defined by $\kappa(X,Y) = \operatorname{tr}(X \circ Y)$ -- a non-degenerate
form on $\mathfrak{gl}(V)$-- remains non-degenerate on the image
$\operatorname{ad}(\operatorname{Lie}(G)) \simeq \operatorname{Lie}(G) \subset \mathfrak{gl}(V)$.

Writing $M$ for the orthogonal complement $M=\operatorname{ad}(\operatorname{Lie}(G))^\perp$ with respect to the form $\kappa$,
we have
$$\mathfrak{gl}(V) = M \oplus \operatorname{ad}(\operatorname{Lie}(G))$$
as $G$-representations.
Write $\pi:\mathfrak{gl}(V) \to \operatorname{ad}(\operatorname{Lie}(G))$ for
the projection on the second factor.

Since $G$ is semisimple, $\operatorname{ad}(\operatorname{Lie}(G)) \subset
\mathfrak{sl}(V)$ so that the identity mapping $I$ satisfies
$\kappa(I,\operatorname{ad}X) = \operatorname{tr}(\operatorname{ad}X) = 0$
for each $X \in \operatorname{Lie}(G)$. Thus $I \in M$.

Write $\lambda$ for the composite mapping
$$G \to \operatorname{GL}(V) \subset \mathfrak{gl}(V) \xrightarrow{\pi} \operatorname{ad}(
\operatorname{Lie}(G))$$.

Since $I \in M$, evidentally $\lambda(1) = 0$. Since $\pi$ is a $G$-module
homomorphism, $\lambda$ is $G$-equivariant.
Moreover, by construction $d\lambda_1$ is the identity mapping. Finally, by $G$-equivariance
the image under $\lambda$ of a maximal torus $T$ is contained in the $T$-fixed
points of $\operatorname{Lie}(G)$, i.e. in $\operatorname{Lie}(T)$.

This verifies the stipulated conditions (1),(2),(3) and (4).

Note that Barsdsley and Richardson go on to show that the restriction of $\lambda$
to the unipotent variety $\mathcal{U} \subset G$ defines a $G$-equivariant isomorphism
$\mathcal{U} \xrightarrow{\sim} \mathcal{N}$ where $\mathcal{N} \subset \operatorname{Lie}(G)$ is the nilpotent variety.

Moreover, by Luna's theorem (a proof valid in positive characteristic is given in Bardsley and Richardson's paper) there are $G$-invariant open subset $U \subset G$ and $U' \subset \operatorname{Lie}(G)$ with $1 \in U$ and $0 \in U'$ such that $\lambda_{\mid U}$ defines a surjective
etale mapping $U \to U'$. So $\lambda$ need not be birational, but it is fairly nice.

Note that Barsdley and Richardson actually formulate the above construction more generally
using representations $V$ of $G$ which are "nice enough" (among other things, the restriction of the traceform on $\mathfrak{gl}(V)$ to the image of $\operatorname{Lie}(G)$ must be non-degenerate), and under suitable assumptions (very roughly: the characteristic should be good for $G$) their construction gives "explicit" Springer isomorphisms $\mathcal{U} \xrightarrow{\sim} \mathcal{N}$ in characteristic $p>0$.