Since this port is practically bleeding at the seems, I figured a thread here on GOG was needed to show people the essential mods that make this port acceptable.

First Off: Make sure you update to 1.01 here on GOG. This fixes and ton of stuff along with adding gamepad support, so make sure that is the first fix you ever apply.

For everything else, here is a post on NeoGaf that covers every fix and texture enhancements that one should apply.
http://www.neogaf.com/forum/showpost.php?p=87939597&postcount=1
[NOTE: Concerning win98 compatibility mode, Also note, I personally find WIN98 mode to down grade the overall frame rate too much to be considered a viable option. Though one should at least try it for themselves once to see if it works for you.]

With that being said, here is a pack I made with every mod that is recommended.
https://www.dropbox.com/s/qgs8rwzn0vjrsco/DP_Fixes.zip?dl=1
This pack includes
1) Durante's DPfix 0.9.5, this fixes the locked 720p resolution issue.
2) InfiniteNine's HD font/HUD/icon pack.
3) Shlomo's improved textures for the in-game menu.
To install, simply extract the contents of this zip into the Deadly Premonition install directory.

I did not include Parsnip's button replacement pack, mentioned in the NeoGaf post, since it is only needed to those that want to use a PS4 controller (or PS2/PS3). If you whish to use it, download it here,
http://beerlake.net/dp-stuff/47549036.dds
and place it in the "tex_override" folder that's inside the "dpfix" folder. This will replace all Xbox360 buttons with PS4 ones.

Also note that the only change I made to the dpfix.ini's default settings was setting it to load custom textures.
If you wish to go messing with the settings, simply open up dpfix.ini with notepad and start editing what you want edited.

Anyways, hope this helps some people here on GOG!

Thanks, sir.

I second TaxAkla's sentiment.

Thanks! I didn't know about HD font and improved textures but I'll apply them all the same as I haven't began to play the game yet, only made it ready to be played.

Changing the compatibility to WIN98 also averts a bunch other possible, lesser problems: sprites not rendering correctly (most notable with the fires around your car at the start of the game), an error with hit-boxes when using melee weapons (making it extremely hard to hit anything with them), not being able to move the guns' aim around freely except for a small section of the screen, in-game lag, and probably a few other things I didn't notice. New players might want to try playing the game both with and without WIN98-compatability to see what works best for them, and look out for these things.

Changing to the HD font also fixes a rare error (which I and a few others' had) that made the in-game text corrupt. You'd notice if you had this error just from the "press enter" screen however.

Also, after installing the DPfix I was unable to boot the game up properly from the Launcher, so if people are still having problems running the game after installing this they might consider trying to just run the DP.exe file directly.

Every time you put the game under compatibility mode it actually make the game runs in single-core mode instead of multi-core (as of patch 1.01). By manually tweaking the Task Manager to make the game utilizing multi-core processor will produce better performance, but still nowhere near as good as the default with no compatibility mode.

I got a problem where the pre-rendered cut-scenes (at least the opening) would freeze at a certain point and didn't notice until recently. Going into msconfig.exe > boot > advanced options, turning on Number of Processors and setting it to 1 fixed it for me. So it might help you too. Windows will ask you to reboot, but I didn't have to. Just remember that you probably want to turn off Number of Processors again when not playing. Edit: This seem to have made some of the other cut-scenes impossible to view at all instead. Frustrating. Edit 2: Now it shows the intro, but not the other aforementioned cut-scenes, even when I force it to use 4 processors as wormholewizards suggested. I officially have no idea what goes on here anymore. Edit 3: I tried it again with 4 processors and now neither worked. So the change I got was probably unrelated to the number of processors. No idea what could actually be causing it. Messing with the amount of processors may solve your problem however. The cut-scenes I can't get to work are 1) talking to the waitress by the rich guy in the wheelchair (forgot his name) at the diner before going to the community center on second day, and 2) going to the community center. Eating at the Sheriff's Office works fine.

The only load time I got any noticeable problems with was the ESC-menu, but after I load it once it works as before for the rest of the session.

Well this did in fact fix my issue with the cut-scenes and the long load times. Though the new problem now is having to exit to the task manager when the game starts. Aside from being inconvenient, DPfix crashes when you alt tab in any form, so going to the task manager is not practical. Is there anyway to make some sort of batch file that starts the exe with all cores?

@painocus
Not sure what your problem is. All cut-scenes play fine for me (as in they don't crash).
Though all these cut-scenes are plane wmc files. This might be a long shot, but is there any issues with your Windows Media Player codecs?

For me it worked by having Task Manager open before running DP.exe, selecting the task manager's window before DP starts properly, setting the cores, and then going to the game's window. No idea if it will work in full-screen mode however.

Edit: This should be what you are looking for: http://www.eightforums.com/tutorials/40339-cpu-affinity-shortcut-program-create-windows.html
Edit 2: I tested it and it didn't seem to work. I think the compatibility mode turned them back off. It was related to the compatibility mode. WIN98 and older has the problem with those cut-scenes, while WIN NT and newer has the gameplay problems. I found out that the game ran smoothest for me on WIN NT mode (everything newer lags a bit and everything older has slightly choppy framerate) so if I can't find a solution I'll just play it on that and switch temporary to WIN98 or 95 if the gameplay requires it.

I'll try updating/repairing my codecs.
Edit: Which made the problem worse, now the intro freezes in the same way, regardless of compatibility mode.
Edit2: I tried a different codec pack and now everything works perfectly! It is strange that it only affected some of the videos before.

The easiest way is by installing third party process manager, i personally recommend ProcessLasso because it is free, light and not bundled with any other unnecessary software.

Well I managed to get it working last night by starting the program and immediately pressing alt tab when it's still loading the logos. I can then go to task manager and set the affinity and go back.
I played for about an hour and while it did fix all the bugs that painocus mentioned, it also degraded the overall frame rate (as you said). So I went back to no compatibility mode at all (I can live with these bugs if it means better frame rate).

I'll update the main post with this info.

Okay, never mind!
I got to a point in the game were not being able to shoot in any direction practically made it impossible!
So I set the game back to win98 mode and made the internal rendering 640x360 so I can get a steady 60fps on WIN98. Bare in mind my PC sucks, so my settings are lower then average.

I sort of got stuck with the modding for this game. The texture files in which readmes specifically tells you to put them in the override folder which is in DPfix folder messed things up for me, the original cutscene started somehow even though I have disabled it. Anyone that has an idea of what I should do? I have renamed the necessary files that said I should do so and enabled texture override.

I have:
DPFix 0.95
HUD Override
Improved Texture

I'm not exactly sure what your talking about.
Are you using the pack on the first post? You should be able to just place the contents of that pack into the main directory of the game and it should work.
What is this disabling the original cutscene?