Make the Budget field visible & accessible to the user: 

1. From Setup click the Object Manager tab, select Opportunity
2. Navigate to Fields & Relationships and open Budget field
3. Select “set field level security” button
4. Make sure that all boxes are checked in the Visible column
5. Only de-select the System Admin in the Read-only column
   
6. Click Save. This allows the Budget field to be editable by end users.

Add Budget field to the Opportunity page layout:

1. From Setup click the Object Manager tab, select Opportunity
2. Navigate to Page Layouts
3. Select page layout, drag & drop to the field to the Opportunity detail section
4. Click Save

Create a rule to populate the budget field with 10% “spend budget” of the Opportunity value:

• From Setup navigate to Workflow Rules, click Continue if the "Understanding Workflow" message appears.  Then click the "New Rule" button

• Create a new rule
• Select the Object you wish to set a budget on and click next. In this example the Opportunity budget.

• Fill out the form like this:
  • Rule name: "Set Budget to 10% of Opp Value"
  • Description: Automatically sets Opp "budget" to 10% of opportunity value so that expenses against this object are drawn-down against budget to help with planning and forecasting spend.
  • Evaluation Criteria: Created & every time it’s edited
  • Rule Criteria: Run this rule if the formula evaluates to true
  • Copy this Formula: ISNEW() || ISCHANGED(Amount)
    

Save

Add Workflow Action named: Update Budget Field

• Unique name: Update_Budget_Field
• Description: Calculate Budget as 10% of Opp value when Opp Amount is new or changed.
• Check the box to: Re-evaluate Workflow Rules after Field Change
• Specify new value Use a Formula to set new value:
• Save => Done =>  Activate.

Check your work: open an Opportunity.  Update the Opp Value & corresponding 10% budget will be assigned.  See if that value is in the budget field. Modify the opp value.  Ensure the budget field updates accordingly to 10% of the new value.

Need more information? Please refer to Salesforce documentation.