Show $H_1(x)$ is partially computable
Many such exercises are best solved by writing down programs in your favorite programming language. However, a typical programming language cannot examine its own source code, or the source code of a subroutine, or of another program, whereas a Turing machine can. So we need to enrich the language with two special constructs:


eval(x,y) which runs source code x on input y and returns the result. This is more or less the universal Turing machine. (Yes, the Python toplevel interactive shell is the same thing as a universal Turing machine.)
uneval(f) which accepts a function f and returns its source code. This allows us to inspect source code, and to simulate it step-by-step (we can implement a "debugger" by hand once we get acces to source code).


Strictly speaking, eval is not needed because it can be implemented in a general-purpose programming language. It amounts to implementing an interpreter for the language inside the language itself, a favorite programming exercise of people who design programmng languages.

So, to so solve your exercise in Python:

def H1(x):
  "Simulate the machine x on input x. Return 1 if it halts, cycle otherwise."
  eval(x, x)
  return 1


That was easy, and to the point. Fiddling with Turing machines just obscures the fact that this is a very easy exercise (although at some point you do need to convince yourself that Turing machines can do all the things that a programming language can do, and vice versa).

By the way, when you pose questions here, you should not assume that everyone knows your textbook by heart. You should explain notation such as $\Phi(x,x)$. I had to guess what you meant.