To unlock the Ultra Trials achievement you must complete 200 trials in Killer Instinct (KI). Trials are various challenges, displayed on the home screen of KI. There are two menus in KI:
1) Meld (this is what is displayed when you first boot up the game; it displays KI news and can be closed or opened by pressing Y);
2) Main Menu (this is to the where you select game modes (single player or multiplayer) or the store.
In KI Season 1, the main menu also listed a set of trials on the homescreen that you could complete to pursue this achievement. In Season 2, the trials menu has been moved to a sub-menu: Single Player > Master > Trials. Go there to see a list of current trials you can complete to earn this achievement.
In addition to counting towards this achievement, completing trials also earn you KP points--the in-game currency used to purchase costumes, accessories, arenas, and so forth--and also contribute to the following three achievements: General Novice, General Apprentice, and General Master (Reach 10, 30, and 60 General Fight Titles).
It's a long journey to complete 200 trials, so it's important to strategize. Things you should know:
1) Displayed Trials: A trial must be displayed in the Trials menu to count towards the total. Four trials are displayed at a time. If you complete a trial, another will take it's place. Generally, if you highlight a trial and press A to select it, it will automatically take you to where you need to go (e.g. selecting 'Perform 4 Shadow Enders in one Vs. Match' will instruct the game to jump you into Vs. match).
2) Skipped Trials: A trial can be temporarily skipped by highlighting it and selecting Y. You can skip up to 9 trials at a time. The number of skips you have remaining is displayed on the bottom left of the home screen, below the Main Menu. So if certain trials are too difficult (e.g. Win 2 consecutive ranked matches with Glacius), don't hesitate to skip it. Skipping it will bring a new trial into the mix that may be easier for you to do. As you complete trials successfully, the Skip count will replenish (i.e., if you skip 3 in a row, the 9 drops to a 6, but as you complete trials, that 6 will turn into a 7, then an 8, and then back to 9). What slowed me down initially was that I was afraid of skipping trials. Don't be. You won't run out of trials. However, while skipped trials always came back for me, some users have reported that easier trials they skipped earlier on never came back. Therefore, you should be very prudent in which trials you skip. Don't skip for the sake of skipping. If a trial is do-able, do it rather than skip it. Save the skips for tougher trials that you are unlikely to complete (e.g. Perform 4 Manuals in one Exhibition Match with Sabrewulf), and are therefore just taking up space on the limited list of trials.
3) Get on a Roll: When you complete a trial usually a similarly themed one will take its place (e.g. beating the 'Play 3 Exhibition Matches with Thunder' will be replaced by 'Play 3 Exhibition Matches with Orchid', etc.). My advice is to find a trial that is easy for you, and then focus on going on a roll with it--that is, beating the subsequent versions of the trial with the other characters. This is a good way to knock out 7 trials at a time.
4) Be a Noob: For some of the trials that pit you against CPU opponents (e.g., Perform 3 consecutive Ultras in Survival mode), you can adjust the difficulty level. You do this by highlighting the trial and then cycling through the difficulty level by pressing LT or RT. The difficulty setting is displayed on the bottom left of the home screen, below the Main Menu. If no difficulty setting is displayed that means that trial doesn't have a difficult setting and you must complete it as is. For those trials that do have a difficulty setting option, the difficulty levels available change depending on the trial, but they usually range from Noob or Beginner on the easy side to Veteran or Champion on the hard side. The default setting is usually Medium, which can actually be difficult enough to be quite frustrating when trying to do certain trials and the CPU is combo breaking you or blocking, or otherwise playing like a stud and dragging out the matches. You should always change it to the easiest setting because it'll allow you to complete the trial sooner and easier. There is no penalty for doing this.
5) Cheat in Practice: Some trials take you to Practice Mode (e.g. Perform a 50-hit Ultra Combo). This is great because it means you can tweak the settings to make it a piece of cake. For instance, if the trial is to Perform a 50-hit Ultra Combo in Practice Mode, select it with A to jump into Practice Mode, then press the Menu button on the controller (the one above the Right Stick) to bring up the Pause Menu. There you can do several things by pressing the Right or Left Bumpers to cycle through the submenus:
a) Dummy Options (change it to 'Stand', which will make the CPU character do nothing)
b) Practice Options (change Health Max to '20%' or '40%', Regenerate Health to 'No', Shadow Meter to 'Infinite', and Instinct Meter to 'Refill', which will allow you to pull off a very long Ultra combo by using endless Shadow Linker attacks to jack up your hit count, initiate the Ultra Combo, use the Instinct Meter to cancel the Ultra Combo, continue the combo by employing more Shadow Linkers until the Combo Limit meter is nearly maxed out, and then re-initiate the Ultra Combo).
6) Cheat in Versus: Gavorkna points out that for trials that take place in Vs. matches, unless it specifically says it needs to be against the CPU, you can do them against another player. So, if you have a second controller, connect it, then, instead of clicking on the trial to hot jump into the trial (which will likely match you against the CPU), manually go to Local Play > Versus > and then choose 'Player vs. Player' so that you can boost the trial by yourself.
7) Double Up: Some trials are their own achievements, such as purchasing skins for characters (e.g. Purchase a skin for Orchid). My advice is to hold off on purchasing skins for the sake of getting an achievement, waiting instead for a 'double up' opportunity in which purchasing the skin nets you both an achievement and a completed trial. This is a good idea not only because character skins cost a lot of KP (the more expensive ones are 5000 KP), but also because some of them are tough to get (e.g. one skin won't unlock for purchase unless you have 50 online ranked victories with that character). So, it's better to save those skins that unlock more easily for a trial completion, rather than jumping the gun and buying it early. If a 'Purchase a skin for X' trial arises before you're ready for it, just skip it (see above) and it'll come back later when you're ready.
8) Respect the Dojo: There are trials for each of the 32 Dojo Lessons. I recommend doing them because completing the Dojo lessons will make you a much better player. That said, some of them are quite hard (looking at you, Lesson 19). While it is okay to skip a Dojo trial to temporarily remove it from the Trial Menu, don't make the mistake of then continuing on in the Dojo (i.e. skipping Dojo Lesson 19 and doing Lesson 20). While you can still go back and beat Dojo Lesson 19, and while the trial for beating Dojo Lesson 19 will also return, everything will be out of whack: i.e. you'll complete Dojo Lesson 19 and it'll say you completed Dojo Lesson 20, even though you already completed that. It's not a huge deal, but it's confusing. To reiterate, though, if you are struggling with a Dojo Lesson, it's fine to skip the trial off the Trial menu list and focus on other trials (because the Dojo lesson trial will return).
Track Your Progress
There are two indicators of your progress.
1) Monster Trials: The first indicator of progress is the sister achievement of Ultra Trials, 'Monster Trials'. Monster Trials requires you to complete 100 trials. Once that pops, you'll know you're half way home.
2) General Fight Titles: The second indicator of progress can be found in the General Fight Titles progress bars. To access this, from the Main Menu go to Single Player > > Fight Archive > Fight Challenges > General > Trials Completed. There it will tell you approximately how many trials you have remaining (e.g. the left side of the bar will say 100, the right side will say 200, and you will see that the meter is about 60% filled, meaning you probably have about 40 trials to go). A caveat on this method for checking progress, or perhaps on this achievement in general, but this achievement popped for me even though the Trials Completed progress bar did not yet hit 200. Also, the progress bars got reset in the update from Season 1 to Season 2. Your actual progress remains there, behind the scenes (so don't worry about that), but the progress bars will be off.
You will generally be notified that you completed a trial at the end of a match or lesson. Once you complete 200 of them, the achievement will also pop at the end of the match. Good luck!