These effects are both triggered abilities that you control, so they will happen sequentially in the order of your choice. Players will have an opportunity to respond before and between their resolutions.
All of the relevant rules for this situation are in the Triggered Abilities section of the rules. First, rule 603.1 says
Triggered abilities have a trigger condition and an effect. They are written as “[Trigger condition], [effect],” and begin with the word “when,” “whenever,” or “at.” They can also be expressed as “[When/Whenever/At] [trigger event], [effect].”
and rule 603.4 says
A triggered ability may read “When/Whenever/At [trigger event], if [condition], [effect].” When the trigger event occurs, the ability checks whether the stated condition is true. The ability triggers only if it is; otherwise it does nothing. If the ability triggers, it checks the stated condition again as it resolves. If the condition isn’t true at that time, the ability is removed from the stack and does nothing. Note that this mirrors the check for legal targets. This rule is referred to as the “intervening ‘if’ clause” rule. (The word “if” has only its normal English meaning anywhere else in the text of a card; this rule only applies to an “if” that immediately follows a trigger condition.)
As you can see, Biovisionary's ability matches this:
At the beginning of the end step, if you control four or more creatures named Biovisionary, you win the game.
In addition, rule 603.7 says
An effect may create a delayed triggered ability that can do something at a later time. A delayed triggered ability will contain “when,” “whenever,” or “at,” although that word won’t usually begin the ability.
And Splinter Twin's ability matches this:
Enchanted creature has "Tap: Create a token that's a copy of this creature. That token has haste. Exile it at the beginning of the next end step."
As you can see, both abilities have the same trigger event: the beginning of the end step. So, they both trigger at the same time. To determine what to do in that situation, we have to look at rule 603.3b:
If multiple abilities have triggered since the last time a player received priority, each player, in APNAP order, puts triggered abilities he or she controls on the stack in any order he or she chooses. (See rule 101.4.) [...]
So, at the beginning of the end step after you activate Splinter Twin, both abilities trigger, and you choose the order in which they are put on the stack. Then they are resolved sequentially from top to bottom, like any other abilities on the stack.
You will win in this situation if you put the Splinter Twin ability on the stack first, then put the Biovisionary ability on the stack. If you do, the situation will play out like this:
You put the Splinter Twin ability on the stack.
You have 4 copies of the Biovisionary copy (each Biovisionary has the ability independently). As you put each on the stack, you check that you have 4 creatures named Biovisionary. You do, so each ability is successfully put on the stack.
Assuming neither player has any responses, the top ability is a Biovisionary ability. As it starts to resolve, you check again that you have 4 creatures named Biovisionary. You still do, so you win.