The latest version needs .NET Framework 4.5 (which comes pre-installed on Windows 8) and AviSynth, so make sure those are installed
I believe avidemux can use avisynth, so I'd recommend you to install avisynth, then download ffms2.dll x86 (only the dll is required) and vsfilter.dll, save the following text in Italics as "something".avs and open it with avidemux. Although I'd use Hybrid instead of avidemux but that's just my personal choice.
TEXT:
LoadPlugin("C:\PATH\TO\ffms2.dll")
LoadPlugin("C:\path\to\VSFilter.dll")
source="C:\PATH\TO\YOUR\VIDEO.mp4"
v=FFVIDEOSOURCE(source).textsub("C:\PATH\TO\YOUR\SUB.SSA")
a=FFAUDIOSOURCE(source)
AV=audiodub(a,v)
return AV
also, use sub station alpha (.ass/.ssa)
Edit: you can skip the audio lines (the ones that start with a and AV) and change return AV to return v if you're only working with the video. But since you're also going to encode the audio you can leave it as it is if you want.
If you have the time and ability to learn it, AviSynth is a great free program. Before you even open it make sure to load in the AVSP mod which adds a ton of great features. If you're not already using it, MeGUI is another free program you can use to deconstruct source videos into raw components and spit out a starter script for AviSynth. After that you can load the script into MeGUI and process it to get the final video.
I can't stress enough how important the AVSpMod is for making AviSynth everything you could need. It adds a ton of functionality and more or less makes it go from being a text only editor to being a fully featured set of tools. If you like using subtitles you can do it from within the program or import a subtitle file from AegisSub with little effort.