Analyze Statement I:
Statement I says: The presence of weaker $\pi$-bonds make alkenes less stable than alkanes.

• An alkene contains a $\mathrm{C=C}$ double bond, which consists of: $1\ \sigma\text{-bond} + 1\ \pi\text{-bond}$
• A $\pi$-bond is weaker than a $\sigma$-bond because it is formed by sideways overlap of orbitals, which is less effective than head-on overlap.
• Because of the weaker $\pi$-bond, alkenes are more reactive and thermodynamically less stable than corresponding alkanes.

Therefore, Statement I is correct.

Analyze Statement II:
Statement II says: The strength of the double bond is greater than that of carbon-carbon single bond.

• A carbon-carbon single bond is one $\sigma$-bond.
• A carbon-carbon double bond has: $1\ \sigma + 1\ \pi$
• Even though the $\pi$-bond is weaker than a $\sigma$-bond, the total bond strength of a double bond is still greater than that of a single bond.
• Typical bond energies are: $E(\mathrm{C-C}) \approx 348\ \text{kJ mol}^{-1}$ $E(\mathrm{C=C}) \approx 614\ \text{kJ mol}^{-1}$
• Hence, a double bond is stronger than a single bond.

Therefore, Statement II is correct.

Conclusion:
Both Statement I and Statement II are correct.

So, the correct option is: $\boxed{\text{A}}$

Comparison with stored answer:
Stored correct answer = A
Derived answer = A
Hence, they agree.