I got it mostly working without a custom msbuild script. Here are the relevant TeamCity build configuration settings:
Artifact paths: %system.teamcity.build.workingDir%\MyProject\obj\Debug\Package\PackageTmp
Type of runner: MSBuild (Runner for MSBuild files)
Build file path: MyProject\MyProject.csproj
Working directory: same as checkout directory
MSBuild version: Microsoft .NET Framework 4.0
MSBuild ToolsVersion: 4.0
Run platform: x86
Targets: Package
Command line parameters to MSBuild.exe: /p:Configuration=Debug
This will compile, package (with web.config transformation), and save the output as artifacts. The only thing missing is copying the output to a specified location, but that could be done either in another TeamCity build configuration with an artifact dependency or with an msbuild script.
Update
Here is an msbuild script that will compile, package (with web.config transformation), and copy the output to my staging server
<?xml version="1.0" encoding="utf-8" ?>
<Project DefaultTargets="Build" xmlns="http://schemas.microsoft.com/developer/msbuild/2003">
<PropertyGroup>
<Configuration Condition=" '$(Configuration)' == '' ">Release</Configuration>
<SolutionName>MySolution</SolutionName>
<SolutionFile>$(SolutionName).sln</SolutionFile>
<ProjectName>MyProject</ProjectName>
<ProjectFile>$(ProjectName)\$(ProjectName).csproj</ProjectFile>
</PropertyGroup>
<Target Name="Build" DependsOnTargets="BuildPackage;CopyOutput" />
<Target Name="BuildPackage">
<MSBuild Projects="$(SolutionFile)" ContinueOnError="false" Targets="Rebuild" Properties="Configuration=$(Configuration)" />
<MSBuild Projects="$(ProjectFile)" ContinueOnError="false" Targets="Package" Properties="Configuration=$(Configuration)" />
</Target>
<Target Name="CopyOutput">
<ItemGroup>
<PackagedFiles Include="$(ProjectName)\obj\$(Configuration)\Package\PackageTmp\**\*.*"/>
</ItemGroup>
<Copy SourceFiles="@(PackagedFiles)" DestinationFiles="@(PackagedFiles->'\\build02\wwwroot\$(ProjectName)\$(Configuration)\%(RecursiveDir)%(Filename)%(Extension)')"/>
</Target>
</Project>
You can also remove the SolutionName and ProjectName properties from the PropertyGroup tag and pass them to msbuild.
msbuild build.xml /p:Configuration=Deploy;SolutionName=MySolution;ProjectName=MyProject