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Let $G$ be a compact Lie group and $\pi:P\to M$ a principal $G$-bundle. Since $G$ is compact, it has an embedding $G\hookrightarrow U(n)$ for some $n$. This embedding determines a unique principal $U(n)$-bundle over $M$ and hence we obtain Chern classes $c_k\in H^{2k}_{\mathrm{dR}}(M;\Bbb{R})$ for $k=1,\ldots,n$ (by applying the Chern-Weil homomorphism to the $U(n)$-invariant polynomials $\mathfrak{u}(n)\to\mathbb{R}$ obtained by expanding the function $\det(\lambda I-\frac{1}{2\pi i}X)$ in powers of $\lambda$).

To what extent do these cohomology classes depend on the choice of embedding $G\hookrightarrow U(n)$?

In other words, if $G\hookrightarrow U(m)$ is another embedding and we construct the respective Chern classes $\tilde{c}_k\in H^{2k}_{\mathrm{dR}}(M;\Bbb R)$, do we have $\tilde{c}_k=c_k$ for $1\leq k\leq\min(m,n)$ and all other classes are zero?

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This certainly depend on the emebding. For instance let $G=U(1)$ the correspondant $C^n$ bundle splits as the sume of line budnles $L^{n_i}$ where the $n_i$ are the weight of the representation, and $L$ is the line bundle associated with the isomorphic representation $U(1)\to U(1)$. Thus the Chern character is the product $\Pi_i^n (1+n_ic)$. Something like this is true for the general case ; the computation reduces the case to the set of irreducible representations of $U(n)$.

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    I do not know the way to associate chern class based on embedding $G\hookrightarrow U(n)$.. Can you give a reference for the same? All I know is, given a vector bundle $E\rightarrow M$ with fibre $\mathbb{C}^r$ and Structure group $Gl(r,\mathbb{C})$, consider Chern-Weil homomorphism $I(G)\rightarrow H^*(M,\mathbb{R})$ and do as said in question... – 2018-12-25
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    Chern classes can be defined either with a n-dimensional vector complex vector bundle, $V$ or a principal $U(n)$ bundle $P$. To a principal $U(n)$ vector bundle $P$ one associates the vector bundle $P\times _{U(n)} \bf C^n$, and conversely, starting with a vector bundle one chooses a hermitian metric and then consider the principal $U(n)$-bundle $Isom (V, \bf C^n)$, the bundle of orthonormal frames of $V$. An embedding $G:\to U$ extends a principal $G$ bundle to a principal $U$ bundle$ – 2018-12-25
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    question asks for a random Lie group $G$ and a principal $G$ bundle $P(M,G)$ you associate a Chern class, right? Or, for a random Lie group $G$ and a Principal $G$ bundle $P(M,G)$ you embed $G$ in some unitary group $U(n)$ and you take the associated $U(n)$ bundle say $Q\rightarrow M$ and for this you associate Chern class..? – 2018-12-25
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    There is no way to define a chern class say for a $O(n)$ bundle. To this end you need to choose an embedding in some $U(n)$, and compute.. – 2018-12-25
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    Ok Ok :) Can you please have a look at https://math.stackexchange.com/questions/3051220/chern-class-cohomology-coefficients-complex-real-integral when you are free.. – 2018-12-25
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    Milnor/Stasheff or Greub/Halperin/VanStone... – 2018-12-25
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    I think Milnor Stasheff does it in topolgical approach, did not see curvature except in appendix... So, I will try that Connections, curvature cohomology book :) Can you please see the question I have mentioned in my comment :) – 2018-12-25
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    Let us [continue this discussion in chat](https://chat.stackexchange.com/rooms/87522/discussion-between-thomas-and-praphulla-koushik). – 2018-12-26