Incremental Cipher

8086 machine code, 70 68 67 bytes

00000000  be 82 00 bf 43 01 57 31  d2 ac 3c 0d 74 2c 89 d1  |....C.W1..<.t,..|
00000010  88 c2 aa e3 f4 4f 28 c1  9f 88 e7 79 02 f6 d9 83  |.....O(....y....|
00000020  f9 0d 9f 76 05 83 e9 1a  f6 d9 30 fc 9e b0 3c 78  |...v......0...<x|
00000030  02 b0 3e f3 aa b0 2a aa  eb cf c6 05 24 b4 09 5a  |..>...*.....$..Z|
00000040  cd 21 c3                                          |.!.|
00000043

How it works:

            |   org 0x100
            |   use16
be 82 00    |       mov si, 0x82        ; source = command line arguments
bf 43 01    |       mov di, result      ; destination = result
57          |       push di
31 d2       |       xor dx, dx          ; clear dx
ac          |   n:  lodsb               ; al = *si++
3c 0d       |       cmp al, 0x0d        ; end of input reached? (newline)
74 2c       |       je q                ; jump to exit in that case
89 d1       |   @@: mov cx, dx          ; store last char in cl
88 c2       |       mov dl, al          ; and store the current char in dl
aa          |       stosb               ; *di++ = al
e3 f4       |       jcxz n              ; skip encoding this char if cx == 0 (only happens for the first char)
4f          |       dec di              ; move di pointer back
28 c1       |       sub cl, al          ; take the difference between this char and the last one
9f          |       lahf                ; store flags from last subtraction in bh
88 e7       |       mov bh, ah
79 02       |       jns @f
f6 d9       |       neg cl              ; make sure cl is positive
83 f9 0d    |   @@: cmp cl, 13          ; which way is shorter?
9f          |       lahf                ; also store these flags
76 05       |       jbe @f
83 e9 1a    |       sub cl, 26          ; invert cl if we're going backwards
f6 d9       |       neg cl
30 fc       |   @@: xor ah, bh          ; xor saved flags together
9e          |       sahf                ; load flags register with the result
b0 3c       |       mov al, '<'
78 02       |       js @f               ; now the sign flag tells us which operator to use
b0 3e       |       mov al, '>'
f3 aa       |   @@: rep stosb           ; while (cx--) *di++ = al
b0 2a       |       mov al, '*'         ; mark the end with an asterisk
aa          |       stosb
eb cf       |       jmp n               ; repeat
c6 05 24    |   q:  mov byte [di], '$'  ; mark end of string
b4 09       |       mov ah, 0x09        ; dos function: print string
5a          |       pop dx              ; dx = string pointer
cd 21       |       int 0x21            ; syscall
c3          |       ret
            |   result rb 0

Python 3, 87 bytes

r,*s=input();p=r
for c in s:d=(ord(p)-ord(c)-13)%26-13;r+='<'*d+'>'*-d+'*';p=c
print(r)

Try it online!

Works with either lowercase or uppercase.

The program builds the output string r as it iterates over the characters in the input string. It stores the previous character as p, and computes the incrementing operation to get from p to the new character c.

The interval between the characters is ord(c)-ord(p), and (ord(c)-ord(p)-13)%26-13 takes it modulo 26 to the interval [-13..12]. A negative result means it's shorter to step down, and a positive result means to step up. This needs to be converted to a string of > or < depending on the sign. Rather than using abs or a conditional, we take advantage of Python's string multiplication s*n giving the empty string when n is negative. In the expression '<'*-d+'>'*d, the wrong-signed part does not contribute.

The initial state is handled by splitting the input into its first character and the rest with Python 3's unpacking r,*s=input(). The initial character is used to start building the string, as well as the initial "previous" char.

Thanks to ovs for suggesting switching to Python 3 to do this unpacking.

Python 3, 110 93 bytes

r,*s=input()
b=r
for a in s:d=(ord(a)-ord(b))%26;r+=['>'*d,'<'*(26-d)][d>13]+'*';b=a
print(r)

Try it online!