Highlight Specific Cell based with condition

I have removed the solved tag for you so that members are aware that you need help.

1. "any cell will be highlight if its next down cell contains same character."

Do you mean contains the same character or do you mean equals the same cell? (Later on in your description you talk about cells being equal not characters). See point 6.

2. e.g. A1="Pen" A2="Pink" Should A1 be highlighted?
3. B1="Pen" B2="Pen" Should B1 be highlighted?
4. A1="Pen" A2="Pen" A3="Pen" Should A2 be highlighted as well since the cell below it (B3) is also the same value?

5. "Example 2: B1=Pen, B2=Paper, B3=Paper, B4=Pencil, B5=Pencil, B6=Paper. Now in this example I want to highlight automatically B2 & B4 as per condition"

Highlight B6 as well presumably ?

6. "Example 1: D1=Computer & D2=Laptop then Both will be highlight because both previous row cell value not match with next row cell value."

This should really be Example 3.
Now you're saying the cell value doesn't match - when you originally described this problem you said "contains same character" See my question in point 1 above.
Why should any of these these be highlighted? This contradicts your statement of "any cell will be highlight if its next down cell contains same character" which we cant tell until you've specified clearly whether we are comparing entire cells or characters within cells.

It looks like you are comparing entire cells not "a character" - You need to be specific, all this ambiguity leads to delays.

Is this any good as a solution?

Select the entire range A1:C6

Conditional Formatting
New Rule
Use a formula to determine...

=COUNTIF(A$1:A$6,A1)>1

Format as required

Still not clear, you keep talking about characters
Do you mean you are comparing complete values or individual single characters?
A1="pen", The word "pen" is a value not a character. The letters "p" "e" and "n" are individual characters that together make up the word (or value) "pen".

Another example

A1="pen"
A2="pound"

A1 is NOT the same value as A2
But both A1 and A2 begin with the same character (the letter "p")
So are you comparing complete values or individual characters?

Also you say highlight when A1 A2 contain the same character but in the B1 B2 example you're highlighting when they are different?
So are we highlighting when things are the same or when things are different? If we are highlighting both are we using the same colour or different colours?