How does the splitting in this molecule take place? Consider the following molecule and data:
  
  
  
  $$
\begin{align}
\mathrm{H_a} &amp;= \pu{1.0 ppm} &amp;\quad J_\mathrm{ab} &amp;= \pu{5.0 Hz}\\
\mathrm{H_b} &amp;= \pu{3.0 ppm} &amp;\quad J_\mathrm{bc} &amp;= \pu{8.0 Hz}\\
\mathrm{H_c} &amp;= \pu{6.0 ppm} &amp;\quad J_\mathrm{ac} &amp;= \pu{1.0 Hz}\\
\end{align}
$$
  
  Draw a splitting diagram for $\mathrm{H_b}.$ $(\pu{1 box} = \pu{1 Hz})$


Tried to draw the splitting diagram and ended up with this diagram:



I am also stuck as to how to draw the spectrum of the whole molecule as I feel $\mathrm{H_a}$ would be triplet of triplet. Am I on the right track and how to draw the sketch of the spectrum?
According to given data, you may consider expected peak patterns are not complicated such as $\mathrm{ABX}$ or $\mathrm{A_2B}$  patterns (more reading look here. This is simply a  $\mathrm{A_2X}$  pattern ($\delta_X = \pu{6 ppm}$ and $\delta_A = \pu{1 ppm}$ without geminal coupling of two $\ce{H_b}$ protons), and hence, peak is a $dt$ for $\ce{H_b}$ as depicted below with relative heights of the peaks:



Similarly, $\ce{H_c}$ is also a $dt$ resonance with $J = 1 \text{ &amp; } \pu{8 Hz}$ and $\ce{H_a}$ is a $tt$ with $J = 1 \text{ &amp; } \pu{5 Hz}$. OP should able to figure out the intensities of relevant peaks. 

A valuable reference:

Roy H. Bible, Jr., Ph.D., In Interpretation of NMR Spectra: An Empirical Approach; Springer Science: New York, NY, 1965 (https://doi.org/10.1007/978-1-4684-8288-1) (Note: Most relavent to this question is chapter 4 of the book titled "Higher-Order Spin Patterns and Multiple Resonance").