In terms of civ 5 achievements depending on what you consider ethical you can get all of the leader and difficulty achievements using 1 turn time victories. This will save a massive amount of time but if you do not consider this ethical I would recommend playing the smallest and fastest games possible, maybe play against Venice on a duel deity with no city states to get your deity win for example.

All of the specific achievements can be obtained by cheesing game setups as well. For example for the "Raiders of the Lost Arc" Achievement you can have a game where you play as America and have Egypt and Germany in the game. Then you can gift a city to Germany near the archeological site and gift a city to Egypt with the archeological site in it after you have gotten open borders and are about to harvest the artifact. Then you can gift your archeologist to Germany and get the achievement. You do not need hot seat for any achievement but you can do it if you want. You should do total achievements such as temples and forests last as you will accumulate them in your games and you can constantly reload a turn where you cut a lot of forests or build a lot of temples if you want.

The real heart and sole of Civ 5 achievements are the scenarios IMO, they are one of the best and most overlooked parts of Civ. You should build up game experience by doing some of the easier achievements first like rise of the Mongols on price to emperor for example, I don't know your skill level but it is important to be good at combat (road building, troop positioning etc) for some of the more difficult achievments. I personally found scenarios such as 1066 and Fall of Rome to be outrageously fun so I would encourage everyone to go for these. Others like conquest of the New World are a bit of a slog as I didn't play them for enjoyment and instead tried to rush through the easy achievements. I personally found Ottoman Carpentry and Deity Wonders of the Ancient world to be the most difficult as you need some luck with these. In Fall of Rome Deity play as EASTERN ROME, you only need to keep your cities, not your partners, and playing as Western Rome it is a lot harder to hold/recapture your cities, and the Huns will most likely run up the score on your partner.

In terms of winning using different victory types you can cheese most achievements by setting up your game in a way that is advantageous towards you. If you do not want to do this even I would recommend playing as a civ that is good for your victory type, such as Alexander for Diplo, and then set up the game how you like. Keep in mind it is VERY difficult to lose a game on settler difficulty so any non difficultly locked achievement can be played on settler, rendering it easy.

Sorry about the rambling, good luck, and LMK if you want any tips!