How do you want to play it?

As they were written.
The Adventure game was conceived of as extra episodes of the show. The Russell T Davies era of the show had websites and much smaller puzzles and games to tie into the episodes. This sequence was what was probably intended by the shows producers.

If you prefer to play them as a straight run.
This is the sequence that makes the most sense if you just want to have as little over lap with the show as possible.

As they were released.
Due to the natural course of game design the games final release dates slipped. It can happen to the best of companies. "Making games is hard" as Valve says. So this is the sequence as the games and the TV show episodes came out.

You simply watch and play the games in sequence. Swapping over at certain points.
I’ve included spoiler text to explain why the games fit where they do.

Starting at season 5 of the show...
"The Eleventh Hour" Steven Moffat
"The Beast Below" Steven Moffat
"Victory of the Daleks" Mark Gatiss

The adventure game EPISODE 1
1 "City of the Daleks" Phil Ford
The Daleks have returned, rebuilt their empire and Amy knows who they are. So it has to happen after Victory of the daleks but as the story is contiguous between here and the removal from Rory from existence then no part of the game can happen until that moment.

Back to the show at episode 4...
4 "The Time of Angels" Steven Moffat
5 "Flesh and Stone" Steven Moffat
6 "The Vampires of Venice" Toby Whithouse
7 "Amy's Choice" Chris Chibnall
8 "The Hungry Earth" Chris Chibnall

The adventure game episode 2
2 "Blood of the Cybermen" Phil Ford

The Show episode 9...
9 "Vincent and the Doctor" Richard Curtis

The adventure game episode 3
3 "TARDIS" James Moran
Amy learns how to fly the Tardis. She is still at the controls at the start of the next TV episode.

The Show episode 10...
10 "The Lodger" Gareth Roberts

The adventure game episode 4.
4 "Shadows of the Vashta Nerada" Phil Ford

The Show episode 11...
11 "The Pandorica Opens" Steven Moffat
There is a continuity issue where Amy does not seem to remember her encounter with the Cybermen. This edit was probably due to the massive delay of the games.
12 "The Big Bang" Steven Moffat

Season 6 of the show....
The continuity is looser here as they had basically given up on releasing the game to fit between episodes of the show. We know because of events inside the game that it is probably after episode 2

Back to the game
5 “The Gunpowder Plot” Phil Ford
Inside the doctors study we can see the Doctor's cot. Which places it before A Good Man Goes to War. The Doctor produces the cot during the TV episode and then leaves before he collects it.

Return to the show at episode 5 and continue as normal.
Episode 5 is where the lead into the Ganger story begins. There is no break where a story can happen between here and leaving the Crib Behind

Season 5 up to...
"The Lodger" Gareth Roberts 12 June 2010

The Game is released...
1 "City of the Daleks" Phil Ford 15 June 2010

Season 5 comes to a close...
"The Pandorica Opens" Steven Moffat 19 June 2010
"The Big Bang" Steven Moffat 26 June 2010

The final episodes of the games first season...
2 "Blood of the Cybermen" Phil Ford 1 July 2010
3 "TARDIS" James Moran 27 August 2010
4 "Shadows of the Vashta Nerada" Phil Ford 22 December 2010

Back to the show...
The 2010 Christmas Special
"A Christmas Carol" Steven Moffat 25 December 2010

Season 6 starts 23 April 2011 and ends in...
"The Wedding of River Song" Steven Moffat 1 October 2011

The game production is announced as having ended and the last episode of the game (intended to be the first episode of the games second season) is released...
5 "The Gunpowder Plot" Phil Ford 31 October 2011

You could just watch season 5 and stop after episode 10 “The Lodger”.

Play the first 4 episodes of the game

Watch the last two episodes of season 5 along with the first two episodes of season 6.

Play the Fifth Episode of the game.